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DESIGN    OF 

POLYPHASE  GENERATORS  AND  MOTORS 


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DESIGN  OF 

POLYPHASE  GENERATORS 
AND  MOTORS 


BY 


HENRY  M.  HOB  ART, 

u 

Consulting   Engineer,   General   Electric   Company;    Member   American   Institute 

Electrical  Engineers;  Institution  Electrical  Engineers;  Institution  Mechanical 

Engineers;  Member  Society  for  the  Promotion  of  Engineering  Education 


McGRAW-HILL   BOOK   COMPANY 

239  WEST  39TH  STREET,  NEW  YORK 

6  BOUVERIE  STREET,  LONDON,  E.  C. 

1913 


Engineering 
Library 


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COPYRIGHT,  1912,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY 


THE. MAPLE . PKKS8- YOUR. PA 


PREFACE 


DURING  several  recent  years  the  author  has  given  courses 
of  lectures  at  three  technical  schools  in  London,  on  the  subject 
of  the  design  of  electric  machinery.  These  three  schools  were: 
the  Northampton  Institute  of  Technology;  Faraday  House; 
and  University  College.  Various  methods  of  procedure  were 
employed  and  these  ultimately  developed  into  a  general  plan 
which  (so  far  as  it  related  to  the  subjects  of  Polyphase  Genera- 
tors and  Polyphase  Motors),  has  been  followed  in  the  present 
treatise.  It  was  the  author's  experience  that  the  students  attend- 
ing his  lectures  took  an  earnest  interest  in  calculating  designs 
of  their  own,  in  parallel  with  the  working  out  of  the  typical 
design  selected  by  the  author  for  the  purpose  of  his  lectures. 
At  the  outset  of  the  course,  each  member  of  the  class  was  assigned 
the  task  of  working  out  a  design  for  a  stipulated  rated  output, 
speed  and  pressure.  Collectively,  the  designs  undertaken  by  the 
class,  constituted  a  series  of  machines,  and  co-operation  was 
encouraged  with  a  view  to  obtaining,  at  the  conclusion  of  the  course 
of  lectures,  a  set  of  consistent  designs.  If  a  student  encountered 
difficulty  or  doubt  concerning  some  feature  of  his  design,  he  was 
encouraged  to  compare  notes  with  the  students  engaged  in  design- 
ing machines  of  the  next  larger  and  smaller  ratings  or  the  next 
higher  and  lower  speeds.  Ultimately  the  results  for  the  entire 
group  of  designs  were  incorporated  in  a  set  of  tables  of  which 
each  student  obtained  blue  prints. 

At  two  of  these  three  colleges,  the  "  sandwich  "  system  was 
in-  operation,  that  is  to  say,  terms  of  attendance  at  the  college 
were  "  sandwiched  "  with  terms  during  which  the  student  was 
employed  in  an  electrical  engineering  works.  The  result  of  the 
author's  opportunities  for  making  comparisons  is  to  the  effect 
that  students  who  were  being  trained  in  accordance  with  the 
"  sandwich  "  system  were  particularly  eager  in  working  out  their 

v 

257829 


vi  PREFACE 

designs.  Their  ambition  to  obtain  knowledge  of  a  practical 
character  had  been  whetted  by  their  early  experiences  of  prac- 
tical work;  they  knew  what  they  wanted  and  they  were  deter- 
mined to  take  full  advantage  of  opportunities  for  obtaining  what 
they  wanted.  One  could  discuss  technical  subjects  with  them 
quite  as  one  would  discuss  them  with  brother  engineers.  There 
was  no  need  to  disguise  difficulties  in  sugar-coated  pills. 

It  will  be  readily  appreciated  that  under  these  circumstances 
there  was  no  necessity  to  devote  time  to  preliminary  dissertations 
on  fundamental  principles.  The  author  did  not  give  these 
lectures  in  the  capacity  of  one  who  is  primarily  a  teacher  but  he 
gave  them  from  the  standpoint  of  an  outside  practitioner  lectur- 
ing on  subjects  with  which  he  had  had  occasion  to  be  especially 
familiar  and  in  which  he  took  a  deep  interest.  It  is  the  function 
of  the  professional  teacher  to  supply  the  student  with  essential 
preparatory  information.  In  the  author's  opinion,  however,  a 
considerable  knowledge  of  fundamental  principles  can  advan- 
tageously be  reviewed  in  ways  which  he  has  employed  in  the 
present  treatise,  namely;  as  occasion  arises  in  the  course  of 
working  out  practical  examples.  Attention  should  be  drawn  to 
the  fact  that  various  important  fundamental  principles  are  there 
stated  and  expounded,  though  without  the  slightest  regard  for 
conventional  methods. 

Of  the  very  large  number  of  college  graduates  who,  from 
time  to  time,  have  worked  under  the  author's  direction  in  connec- 
tion with  the  design  of  dynamo-electric  machinery,  instances 
have  been  rare  where  the  graduate  has  possessed  any  consider- 
able amount  of  useful  knowledge  of  the  subject.  Such  knowledge 
as  he  has  possessed  on  leaving  college,  has  been  of  a  theoretical 
character.  Furthermore,  it  has  been  exceedingly  vague  and  poorly 
assimilated,  and  it  has  not  been  of  the  slightest  use  in  practical 
designing  work.  One  is  forced  to  the  conclusion  that  teachers 
are  lecturing  completely  over  the  heads  of  their  students  far  more 
often  than  is  generally  realized.  The  author  is  of  the  opinion 
that  the  most  effective  way  to  teach  the  design  of  electric 
machinery  is  to  lead  the  student,  step  by  step,  through  the  actual 
calculations,  simultaneously  requiring  him  to  work  out  designs 
by  himself  and  insisting  that  he  shall  go  over  each  design  again 
and  again,  only  arriving  at  the  final  design  as  the  result  of  a  study 
of  many  alternatives.  It  is  only  by  making  shoes  that  one  learns 


PREFACE  vii 

to  be  a  shoemaker  and  it  is  only  by  designing  that  one  learns 
to  be  a  designer. 

While  the  author  has  not  hesitated  to  incorporate  in  his  treatise, 
aspects  of  the  subject  which  are  of  an  advanced  character  and  of 
considerable  difficulty,  he  wishes  to  disclaim  explicitly  any  pro- 
fession to  comprehensively  covering  the  entire  ground.  The 
professional  designer  of  polyphase  generators  gives  exhaustive 
consideration  to  many  difficult  matters  other  than  those  dis- 
cussed in  the  present  treatise.  He  must,  for  example,  make 
careful  provision  for  the  enormous  mechanical  stresses  occurring 
in  the  end  connections  on  the  occasions  of  short  circuits  on  the 
system  supplied  from  the  generators,  and  when  a  generator  is 
thrown  on  the  circuit  with  insufficient  attention  to  synchronizing. 
He  will  often  have  to  take  into  account,  in  laying  out  the  design, 
that  the  proportions  shall  be  such  as  to  ensure  satisfactory  opera- 
tion in  parallel  with  other  generators  already  in  service,  and  it 
will,  furthermore,  be  necessary  to  modify  the  design  to  suit  it 
to  the  characteristics  of  the  prime  mover  from  which  it  will  be 
driven.  This  will  involve  complicated  questions  relating  to 
permissible  angular  variations  from  uniform  speed  of  rotation. 
The  designer  of  high-pressure  polyphase  generators  will  find  it 
necessary  to  acquire  a  thorough  knowledge  of  the  properties  of 
insulating  materials;  of  the  laws  of  the  flow  of  heat  through  them; 
of  their  gradual  deterioration  under  the  influence  of  prolonged 
subjection  to  high  temperatures  and  to  corona  influences.  He 
must  also  study  the  effects  on  insulating  materials  of  minute 
traces  of  acids  and  of  the  presence  of  moisture;  and,  in  general, 
the  ageing  of  insulation  from  all  manner  of  causes.  The  designer 
of  polyphase  generators  can  employ  to  excellent  advantage  a 
thorough  knowledge  of  the  design  of  fans;  in  fact,  a  large  part 
of  his  attention  will  require  to  be  given  to  calculations  relating 
to  the  flow  of  air  through  passages  of  various  kinds  and  under 
various  conditions.  The  prevention  of  noise  in  the  operation 
of  machines  which  are  cooled  by  air  is  a  matter  requiring  much 
study.  There  occur  in  all  generators  losses  of  a  more  or  less 
obscure  nature,  appropriately  termed  "  stray "  losses,  and  a 
wide  experience  in  design  is  essential  to  minimizing  such  losses 
and  thereby  obtaining  minimum  temperature  rise  and  maximum 
efficiency.  Furthermore,  mention  should  be  made  of  the  import- 
ance of  providing  a  design  for  appropriate  wave  shape  and  insur- 


viii  PREFACE 

ing  the  absence  of  objectionable  harmonics.  To  deal  compre- 
hensively with  these  and  other  related  matters  a  very  extensive 
treatise  would  be  necessary. 

While  it  will  now  be  evident  that  the  designer  must  extend 
his  studies  beyond  the  limits  of  the  present  treatise  in  acquaint- 
ing himself  with  the  important  subjects  above  mentioned,  the 
familiarity  with  the  subject  of  the  design  of  polyphase  generators 
and  motors  which"  can  be  acquired  by  a  study  of  the  present 
treatise,  will  nevertheless  be  of  a  decidedly  advanced  character. 
It  can  be  best  amplified  to  the  necessary  extent  when  the  problems 
arise  in  the  course  of  the  designer's  professional  work,  by  con- 
sulting papers  and  discussions  published  in  the  Proceedings  of 
Electrical  Engineering  Societies. 

Just  as  the  design  of  machinery  for  continuous  electricity 
crystallizes  around  the  design  of  the  commutator  as  a  nucleus, 
so  in  the  design  of  polyphase  generators,  a  discussion  of  the  pre- 
determination of  the  field  excitation  under  various  conditions 
of  load  as  regards  amount  and  phase,  serves  as  a  basis  for  acquiring 
familiarity  with  the  properties  of  machines  of  this  class.  The 
author  has  taken  the  opportunity  of  presenting  a  method  of 
dealing  with  the  subject  of  the  predetermination  of  the  required 
excitation  for  specified  loads,  which  in  his  opinion  conforms  more 
closely  with  the  actual  occurrences  than  is  the  case  with  any 
other  method  with  which  he  is  acquainted.  In  dealing  with  the 
design  of  polyphase  induction  motors,  the  calculations  crystallize 
out  around  the  circle  ratio,  and  this  may  be  considered  the  nucleus 
for  the  design.  The  insight  as  regards  the  actual  occurrences 
in  an  induction  motor  which  may  be  acquired  by  accustoming 
one's  self  to  construct  mentally  its  circle  diagram,  should,  in  the 
author's  opinion,  justify  a  much  wider  use  of  the  circle  diagram 
than  is  at  present  the  case  in  America. 

The  author  desires  to  acknowledge  the  courtesy  of  the  Editor 
of  the  General  Electric  Review  for  permission  to  employ  in  Chap- 
ter VI,  certain  portions  of  articles  which  the  author  first  published 
in  the  columns  of  that  journal,  and  to.  Mr.  P.  R.  Fortin  for 
assistance  in  the  preparation  of  the  illustrations. 

HENRY  M.  HOBART,  M.Inst.C.E. 
November,  1912. 


CONTENTS 


PAGE 

PREFACE .• .  . .     v 

CHAPTER  I 

INTRODUCTION 1 

CHAPTER  II 

CALCULATIONS  FOR  A  2500-KVA.  THREE-PHASE  SALIENT  POLE  GENERATOR      3 

CHAPTER  III 

POLYPHASE  GENERATORS  WITH  DISTRIBUTED  FIELD  WINDINGS 99 

CHAPTER  IV 

THE  DESIGN  OF  A  POLYPHASE  INDUCTION  MOTOR  WITH  A  SQUIRREL- 
CAGE  ROTOR  105 

CHAPTER  V 
SLIP-RING  INDUCTION  MOTORS 195 

CHAPTER  VI 

SYNCHRONOUS  MOTORS  VERSUS  INDUCTION  MOTORS 202 

CHAPTER  VII 

THE  INDUCTION  GENERATOR 213 

CHAPTER    VIII 

EXAMPLES  FOR  PRACTICE  IN  DESIGNING  POLYPHASE  GENERATORS  AND 

MOTORS 225 

APPENDIX      1 247 

APPENDIX    II 252 

APPENDIX  III 256 

APPENDIX  IV 257 

INDEX 259 

ix 


DESIGN  OF  POLYPHASE  GENERATORS 
AND  MOTORS 


CHAPTER  I 
INTRODUCTORY 

POLYPHASE  generators  may  be  of  either  the  synchronous  or 
the  induction  type.  Whereas  tens  of  thousands  of  synchronous 
generators  are  in  operation,  only  very  few  generators  of  the  induc- 
tion type  have  ever  been  built.  In  the  case  of  polyphase  motors, 
however,  while  hundreds  of  thousands  of  the  induction  type  are 
in  operation,  the  synchronous  type  is  still  comparatively  seldom 
employed.  Thus  it  is  in  accord  with  the  relative  importance  of 
the  respective  types  that  the  greater  part  of  this  treatise  is  devoted 
to  setting  forth  methods  of  designing  synchronous  generators 
and  induction  motors.  Brief  chapters  are,  however,  devoted  to 
the  other  two  types,  namely  induction  generators  and  synchronous 
motors.  The  design  of  an  induction  generator  involves  consider- 
ations closely  similar  to  those  relating  to  the  design  of  an  induc- 
tion motor.  In  fact,  a  machine  designed  for  operation  as  an  induc- 
tion motor  will  usually  give  an  excellent  performance  when 
employed  as  an  induction  generator.  Similarly,  while  certain 
modifications  are  required  to  obtain  the  best  results,  a  machine 
designed  for  operation  as  a  synchronous  generator  will  usually 
be  suitable  for  operation  as  a  synchronous  motor. 

In  dealing  with  the  two  chief  types:  the  author  has  adopted 
the  plan  of  taking  up  immediately  the  calculations  .for  a  simple 
design  with  a  given  rating  as  regards  output,  speed,  periodicity 
and  pressure.  In  the  course  of  these  calculations,  a  reasonable 
amount  of  familiarity  will  be  acquired  with  the  leading  principles 
involved,  and  the  reader  will  be  prepared  profitably  to  consult 


2  POLYPHASE  GENERATORS  AND  MOTORS 

advanced  treatises  dealing  with  the  many  refinements  essential 
to  success  in  designing.  A  sound  knowledge  of  these  refinements 
can  only  be  acquired  in  the  course  of  the  practice  of  designing  as 
a  profession.  The  subject  of  the  design  of  dynamo  electric  machin- 
ery is  being  continually  discussed  from  various  viewpoints  in  the 
papers  contributed  to  the  Proceedings  of  engineering  societies. 
Attempts  have  been  made  to  correlate  in  treatises  the  entire 
accumulation  of  present  knowledge  relating  to  the  design  of  dyna- 
mo-electric machinery,  but  such  treatises  necessarily  extend 
into  several  volumes  and  even  then  valuable  fundamental  outlines 
of  the  subject  are  apt  to  be  obscured  by  the  mass  of  details. 
No  such  attempt  is  made  in  the  present  instance;  the  discussion 
is  restricted  to  the  fundamental  outlines. 

There  are  given  in  Appendices  1  and  2,  bibliographies  of 
papers  contributed  to  the  Journal  of  the  Institution  of  Electrical 
Engineers  and  to  the  Transactions  of  the  American  Institute  of 
Electrical  Engineers.  These  papers  deal  with  many  important 
matters  of  which  advanced  designers  must  have  a  thorough 
knowledge.  After  acquiring  proficiency  in  carrying  through  the 
fundamental  calculations  with  which  the  present  treatise  deals, 
a  study  of  these  papers  will  be  profitable.  Indeed  such  extended 
study  is  essential  to  those  engineers  who  propose  to  adopt  designing 
as  a  profession. 


CHAPTER  II 

CALCULATIONS  FOR  A  2500-KVA.  THREE-PHASE  SALIENT  POLE 

GENERATOR 

LET  us  at  once  proceed  with  the  calculation  of  a  design  for  a 
three-phase  generator.  Let  its  rated  output  be  2500  kilo  olt 
amperes  at  a  power  factor  of  0.90.  Let  its  speed  be  375  revolutions 
per  minute  and  let  it  be  required  to  provide  25-cycle  electricity. 
Let  it  be  further  required  that  the  generator  shall  provide  a  ter- 
minal pressure  of  12  000  volts.  We  shall  equip  the  machine 
with  a  Y-connected  stator  winding.  The  phase  pressure  will  be: 

12000     12000 


The  Number  of  Poles.     Since  the  machine  is  driven  at  a  speed 
of  (-  —  =)6.25  revolutions  per  second,   and  since  the  required 


,60 

periodicity  is  25  cycles  per  second,  it  follows  that  we  must  arrange 
for: 

25      \ 

r-  =  )  4.0  cycles  per  revolution, 
.^o     / 

Consequently  we  must  provide  four  pairs  of  poles,  or  (2X4  =  )8 
poles. 

Denoting  by  P  the  number  of  poles,  by  R  the  speed  in  revolu- 
tions per  minute,  and  by  ~  the  periodicity  in  cycles  per  second, 
we  have  the  formula: 


R 

In  this  treatise  the  power  factor  will  be  denoted   by  G.     Since 
for  our  machine  the  rated  load  is  2500  kva.  for  (7  =  0.90,  we  may 

3 


4  POLYPHASE  GENERATORS  AND  MOTORS 

also  say  that  the  design  is  for  a  rated  output  of  (0.90X2500  =  )  2250 
kw.  at  a  power  factor  of  0.90. 

So  many  alternators  have  been  built  and  analyzed  that  the 
design  of  a  machine  for  any  particular  rating  is  no  longer  a  matter 
which  should  be  undertaken  without  any  reference  to  accum- 
ulated experience.  From  experience  with  many  machines,  design- 
ers have  arrived  at  data  from  which  they  can  obtain  in  advance 
some  rough  idea  of  the  proportions  which  will  be  most  appro- 
priate. It  is  not  to  be  concluded  that  the  designing  of  a  machine 
by  reference  to  these  data  is  a  matter  of  mere  routine  copying. 
On  the  contrary,  even  by  making  use  of  all  the  data  available, 
there  is  ample  opportunity  for  the  exercise  of  judgment  and  origin- 
ality in  arriving  at  the  particular  design  required. 

The  Air-gap  Diameter.  Let  us  denote  by  D  the  internal 
diameter  of  the  stator.  Usually  we  shall  express  D  in  centimeters 
(cm.),  but  occasionally  it  will  be  more  convenient  to  express  it 
in  millimeters  (mm.),  and  also  occasionally  in  decimeters  (dm.), 
and  in  meters  (m). 

Since  the  internal  diameter  of  the  stator  is  but  slightly  in  excess 
of  the  external  diameter  of  the  rotor,  it  is  often  convenient 
briefly  to  describe  D  as  the  "  air-gap  diameter,"  but  we  must  not 
forget  that  strictly  speaking,  it  is  the  internal  diameter  of  the 
stator  and  is  consequently  a  little  greater  than  the  external 
diameter  of  the  rotor. 

The  Polar  Pitch.  Let  us  further  denote  the  polar  pitch, 
(in  cm.),  by  T.  By  the  polar  pitch  is  meant  the  distance,  measured 
at  the  inner  circumference  of  the  stator,  from  the  center  of  one 
pole  to  the  center  of  the  next  adjacent  pole.  The  polar  pitch  T 
is  the  very  first  dimension  for  which  we  wish  to  derive  a  rough 
preliminary  value.  *  The  values  of  T  given  in  Table  1  are  indicated 
by  experience  to  be  good  preliminary  values  for  designs  of  the 
numbers  of  poles,  the  periodicity  and  the  output  shown  against 
them  in  the  Table. 

The  table  indicates  that  the  polar  pitch  T  should  have  a  value 
of  70  cm. 


Since  the  machine  has  8  poles,  the  internal  periphery  of  the 
stator  is  : 

8X70  =  560  cm. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR 


TABLE  1. — VALUES  OF  T  AND  £. 


P=4. 

P=6. 

P=8. 

P  =  12 

R  = 

R  = 

R  = 

R  = 

Rated  Output  in 
Kva. 

/    750  for  25  ^ 
\1500for50  — 

/    500  for  25- 
1  1000  for  50  — 

/  375  for  25  — 
\  750  for  50  - 

1  250  for  25  — 
\  500  for  50  — 

T 

£ 

r 

« 

T 

« 

T 

1 

r      250 

56 

1.12 

48 

1.15 

44 

1.20 

38 

1.22 

500 

68 

1.25 

58 

1.30 

50 

1.35 

43 

1.40 

o> 

1000 

78 

1.38 

65 

1.45 

57 

1.55 

50 

1.65 

"3 

2000 

92 

1.45 

77 

1.55 

67 

1.75 

60 

1.90 

10 

4000 

117 

1.55 

97 

1.70 

90 

1.85 

82 

2.00 

<M 

6000 

141 

1.45 

126 

1.65 

115 

1.70 

106 

2.00 

i  8000 

163 

1.35 

148 

1.55 

135 

1.60 

130 

1.85 

r    250 

46 

1.04 

40 

.15 

36 

1.20 

32 

1.10 

$ 

500 

53 

1.11 

46 

.30 

41 

1.35 

38 

1.25 

'S 

1000 

64 

1.18 

55 

.45 

49 

1.45 

49 

1.50 

0 

2000 

75 

1.26 

65 

.50 

57 

1.60 

51 

1.70 

8 

4000 

82 

1.26 

75 

.60 

68 

1.70 

60 

1.78 

6000 

82 

1.17 

82 

.55 

73 

1.60 

66 

1.85 

P  =  16. 

P=24. 

P=32. 

R  = 

P=48. 

P=64. 

Rated 
Output  in 
Kva. 

f  187.  5  for  25  — 
\  375  for  50  ^ 

I  125  for  25  — 
\  250  for  50  ^ 

J93.5  for  25- 
1  187.5  for  50- 

R  = 

=  125  for  50  - 

R  = 
=  93.5for50<- 

T 

i 

r 

« 

T 

e 

T 

i 

T 

« 

250 

34 

1.24 

31 

1.28 

30 

1.35 

02 

0 

500 

39 

1.44 

35 

1.50 

32 

1.55 

fl 

1000 

45 

1.70 

40 

1.75 

37 

1.80 

o 

2000 

56 

2.00 

50 

2.10 

44 

2.20 

CN 

4000 

76 

2.10 

70 

2.30 

62 

2.40 

6000 

100 

2.15 

250 

30 

1.18 

26 

1.20 

25 

1.25 

25 

1.30 

24 

1.35 

i 

500 

34 

1.30 

31 

1.35 

28 

1.40 

27 

1.45 

26 

1.50 

1- 

1000 

40 

1.45 

36 

1.50 

32 

1.56 

30 

1.65 

28 

1.75 

O 

2000 

46 

1.65 

41 

1.70 

37 

1.80 

34 

1.90 

31 

2.05 

s 

4000 

53 

1.85 

48 

2.00 

44 

2.10 

39 

2.30 

35 

2.45 

6000 

60 

1.90 

54 

2.00 

49 

2.15 

6  POLYPHASE  GENERATORS  AND  MOTORS 

Consequently  for  D,  the  diameter  at  the  air  gap,  we  have  : 


X 

In  general,  we  have  the  relation 


The  Output  Coefficient.  It  will  be  seen  that  in  Table  1,  each 
value  of  T  is  accompanied  by  a  value  designated  as  £.  This 
quantity  is  termed  the  "  output  coefficient  "  and  its  general 
nature  may  be  explained  by  reference  to  the  following  formula: 

..  _  Output  in  volt  amperes 

(Dia.  in  decimeter)2  X  (Gross  core  length  in  dm.)  XR' 

In  Table  1,  the  appropriate  value  for  £  is  given  as  1.78. 
$-1.78. 

Transposing  the  output-coefficient  formula  and  substituting 
all  the  known  quantities,  we  have: 

2  500  000 
Gross  core  length  (in  dm.)  =  17  32x375x1  73 

=  11.8  dm. 
=  118  cm. 

The  term  \g  is  employed  to  denote  the  gross  core  length  in  cm. 
Thus  we  have  : 


D  and  \g  are  two  of  the  most  characteristic  dimensions  of  the 
design,  and  indicate  respectively  the  air-gap  diameter  and  the 
gross  core  length  of  the  stator  core. 

It  is  not  to  be  concluded  that  we  shall  necessarily,  in  the  com- 
pleted design,  adhere  to  the  precise  values  originally  assigned 
to  these  dimensions.  On  the  contrary,  these  preliminary  values 
simply  constitute  starting  points  from  which  to  proceed  until 
the  design  is  sufficiently  advanced  to  ascertain  whether  modified 
dimensions  would  be  more  suitable. 

Discussion  of  the  Significance  of  the  Output  Coefficient 
Formula.  Let  us  consider  the  constitution  of  the  formula  above 


CALCULATIONS  FOR  2500-KVA.  GENERATOR 


given  for  ?  the  output  coefficient.     Two  of  the  terms  of  the  denom- 
inator are  D2  and  \g.     Their  product,  or  D2\g,  is  of  the  nature  of 

a  volume.  If  we  were  to  multiply  it  by  —  we  should  have  the  vol- 
ume of  a  cylinder  with  a  diameter  D  and  a  length  \g.  But  leav- 
ing out  the  constant  -  does  not  alter  the  nature  of  the  expression 
It  is  still  of  the  nature  of  a  volume.  Thus: 


t  _     Output  in  volt  amperes 
(A  volumetric  quantity)  XR' 

Therefore,  £  =  output  per  unit  of  a  volumetric  quantity 
(D2\g)  and  per  revolution  per  minute  (R).  In  other  words,  ? 
is  the  output  per  unit  of  D2\g  per  revolution  per  minute. 

Obviously  the  greater  the  value  of  £,  the  more  are  we  obtaining 
from  every  unit  of  volume.  It  is  now  easy  to  understand  that 
in  the  interests  of  obtaining  as  small  and  low-priced  a  machine 
as  is  consistent  with  good  quality,  we  must  strive  to  employ  the 
highest  practicable  value  of  ?,  the  output  coefficient. 

While  in  most  of  our  calculations  we  shall  express  D  and  ~kg 
in  centimeters,  it  will  occasionally  be  more  convenient,  on  account 
of  the  magnitude  of  the  results  involved,  to  express  these  quan- 
tities in  decimeters  (as  in  the  output  coefficient  formula),  and, 
in  other  instances,  in  meters.  In  Table  2,  are  given  some  rough 
values  connecting  the  total  net  weight  with  D2\g. 

TABLE  2.  —  RELATION  BETWEEN  D*\g  AND  TOTAL  NET  WEIGHT. 


D*\g  (D  and  \g  in 
meters)  . 

Total  Net  Weight  in  Tons. 

Low  Speed. 

High  Speed. 

1 

26 

30 

2 

40 

50 

3 

50 

65 

5 

68 

85 

7 

80 

105 

10 

90 

125 

15 

108 

150 

20 

120 

25 

130 

30 

135 

40 

143 

8 


POLYPHASE  GENERATORS  AND  MOTORS 


The  Peripheral  Loading.  Having  determined  upon  the  per- 
iphery (xXD  or  PXT),  the  next  point  to  be  decided  relates  to  the 
so-called  "  peripheral  loading."  The  peripheral  loading  may  be 
defined  as  the  ampere-conductors  per  centimeter  of  air-gap 
periphery  at  the  rated  load  of  the  machine.  The  range  of  appro- 
priate values  has  been  arrived  at  by  experience  with  successive 
designs.  Such  values  are  given  in  Table  3.  They  are  only  rough 
indications, 'but  they  are  of  assistance  in  the  preliminary  stages 
of  the  preparation  of  a  design. 

TABLE  3. — THE  PERIPHERAL  LOADING. 


Rated 
Output 
(in  kva.). 

Peripheral  Loading,  in  Ampere  Conductors  per  cm.  of  Periphery  at  the  Air  Gap. 

25  Cycles. 

50  Cycles. 

P=4. 

P=8. 

P=16. 

P=32. 

P=4. 

P=8. 

P=16. 

P=32. 

P=64. 

500 

160 

200 

220 

240 

140 

160 

1£0 

200 

210 

1000 

210 

240 

260 

270 

170 

200 

210 

220 

230 

5000 

230 

260 

290 

300 

210 

220 

230 

240 

250 

10000 

240 

280 

310 

320 

220 

240 

250 

260 

270 

For  our  8-pole,  25-cycle,  2500-kva  machine  we  see  that  the 
values  should  be  of  the  order  of:  250  ampere-conductors  per 
cm.  of  periphery. 

Since  the  polar  pitch,  T,  is  70  cm.  we  should  provide  about: 
250X70  =  17500  ampere-conductors  per  pole.  Our  machine  is 
a  three  phaser.  Consequently: 

Ampere-conductors  per  pole  per  phase  =  —      —  =  5830. 


The  Current  at  the  Rated  Load.  The  ampere-conductors 
are  the  product  of  the  current  at  rated  load,  7,  and  the  number 
of  conductors.  For  our  2500-kva.,  12  000-volt  machine,  we  have: 

12  000 
Phase  pressure  =  — -j=-  =6950  volts; 

2  500  000 
Output  per  phase  =  —  —  =833  000  volt-amperes; 

o 

833  000 


CALCULATIONS  FOR  2500-KVA.  GENERATOR 


We  can  now  ascertain  the  number  of  conductors  per  pole  per 
phase  by  dividing  the  ampere-conductors  by  the  amperes,  as 
follows : 


^     A  583° 

Conductors  =  — — 


48.6. 


This  result  must  be  rounded  off  to  some  value  which  will  be 
divisible  by  the  number  of  slots  per  pole  per  phase. 

The  Number  of  Slots.  The  performance  of  a  machine  is  in 
most  respects  more  satisfactory  the  greater  the  number  of  slots 
per  pole  per  phase.  But  considerations  of  insulation  impose 
limitations.  For  12  000-volt  generators  we  may  obtain  from 
Table  4,  reasonable  preliminary  assumptions  for  the  number  of 
slots  per  pole  per  phase,  for  three-phase  generators,  as  a  function 
of  T,  the  polar  pitch: 

TABLE  4. — RELATION  BETWEEN  T  AND  THE  NUMBER  OF  SLOTS  PER  POLE 

PER  PHASE. 


T  the  Polar  Pitch  in  cm. 

Number  of  Slots  per  Pole 
per  Phase. 

20 
25 
30 

2 
3 
3 

35 
40 
50 

3 

4 

4 

60 
80 
100 

5 
5 
6 

120 

7 

Let  us  take  for  our  design,  5  slots  per  pole  per  phase.  Since 
we  want  some  48.6  conductors  per  pole  per  phase,  let  us  take 
10  conductors  per  slot  and  round  off  the  number  of  conductors 
per  pole  per  phase,  to: 

5X10  =  50. 

The  Current  Density.  Our  alternator  is  for  12  000  volts. 
At  this  pressure  the  thickness  of  the  insulation  required  is  so 


10 


POLYPHASE  GENERATORS  AND  MOTORS 


considerable  as  to  be  a  serious  impediment  to  the  escape  of  heat 
from  the  armature  conductors.  Consequently  it  is  necessary 
to  proportion  these  conductors  for  a  lower  current  density  than 
would  be  necessary  for  a  machine  for  lower  pressure.  A  current 
density  of  some  3.0  amperes  per  sq.mm.  will  be  suitable.  The 
current  per  conductor  is  120  amperes.  Consequently  each  con- 
ductor must  have  a  cross-section  of  some: 


120 


=  40  sq.mm. 


Let  us  make  each  conductor  12  mm.  wide  by  3.3  mm.  high  and 
let  us  arrange  above  each  other  the  ten  conductors  which  occupy 
each  slot.  Each  conductor  will  be  insulated  by  impregnated 
braid  to  a  depth  of  0.3  mm.  Consequently  the  dimensions  over 
the  braid  will  be: 

12.6X3.9. 

The  ten  conductors  will  thus  occupy  a  space  12.6  mm.  wide  by 

10X3.9  =  39.0  mm.  high. 

The  Slot  Insulation.  Outside  of  this  group  of  conductors 
comes  the  slot  lining.  In  Table  5  are  given  suitable  values  for 
the  thickness  of  the  slot  lining  for  machines  for  various  pressures  : 

TABLE  5.  —  THICKNESS  OF  SLOT  INSULATION. 


Normal  Pressure  in  Volts. 

Thickness  of  Slot  Lining 
in  mm. 

500 
1000 
2000 

0.9 
1.4 

2.3 

3000 
4000 
6000 

2.9 
3.3 
4.0 

8000 
10000 
12000 

4.7 
5.2 
5.6 

CALCULATIONS  FOR  2500-KVA.  GENERATOR       11 


For  our  alternator  the  slot  insulation  should  have  a  thickness 
of  5.6  mm.  all  round  the  group  of  conductors.  The  dimensions 
to  the  outside  of  the  insulation  will  thus  be: 


2X5.6+12.6  =  11.2+12.6  =  23.8  mm., 


by: 


2X5.6+39.0  =  11.2+39.0  =  50.2  mm. 
Thus  the  insulated  dimensions  are: 
23.8X50.2. 

The  Dimensions  of  the  Stator  Slot.  Allowing  7  mm.  at  the 
top  of  the  slot  for  a  retaining 
wedge  and  allowing  0.3  mm.  addi- 
tional width  for  tolerence  in 
assembling  the  slotted  punchings, 
we  arrive  at  the  following  dimen- 
sions for  the  stator  slot : 


•-24.1mm — 


Depth  =  50.2 +7     =57.2  mm.; 
Width  =  23.8+0.3  =  24.1  mm. 

The  above  design  for  the  slot 
is  shown  in  Fig.  1.  This  can 
only  be  regarded  as  a  prelimin- 
ary design.  At  a  later  stage  of 
the  calculations,  various  consid- 
erations, such  as  tooth  density, 
may  require  that  the  design  be  modified. 

The  Slot  Space  Factor.     The  gross  area  of  the  slot  is: 

57.2X24.1  =  1380  sq.mm. 
Thus  the  copper  only  occupies: 

=  29.0  per  cent. 


L.1 


t 

FIG.  1.— Stator  Slot  for  12  000-volt, 
2500-kva.,  375  r.p.m.,  25-cycle, 
Three-phase  Generator. 


of  the  total  available  space. 


12         POLYPHASE  GENERATORS  AND  MOTORS 

We  express  this  by  stating  that  the  "  slot  space  factor  "  is 
equal  to  0.29.  The  slot  space  factor  is  lower  the  higher  the  work- 
ing pressure  for  which  the  alternator  is  designed. 

THE  MAGNETIC  FLUX 

Having  now  made  provision  for  the  armature  windings,  we 
must  proceed  to  provide  for  the  magnetic  flux  with  which  these 
windings  are  linked.  Let  us  start  out  from  fundamental  con- 
siderations. 

Dynamic  Induction.  A  conductor  of  1  cm.  length  is  moved 
in  a  direction  normal  to  its  length  and  normal  to  the  direction  of  a 
magnetic  field,  of  which  the  density  is  1  line  per  square  centimeter. 
Let  the  velocity  with  which  this  conductor  is  moved,  be  1  cm.  per 
second.  Then  the  pressure  set  up  between  the  two  ends  of  this 
conductor  will  be  1 X 1 X 1 X  10~8  volt.  If  the  conductor,  instead 
of  being  of  1  cm.  length,  has  a  length  of  10  cm.  then  the  pressure 
between  the  ends  becomes 

10  X 1 XI  X10~8  volts. 

If  the  field  density  instead  of  being  1  line  per  sq.cm.,  is  10  000 
lines  per  sq.cm.,  then  the  pressure  will  be 

10  X 10  000  X 1 X  10~8  volts. 

If  the  speed  of  the  conductor  is  1000  cm.  per  second  instead  of 
1  cm.  per  second,  then  the  pressure  will  be 

10  X 10  000  X 1000  X  10~8  volts. 

Obviously  with  our  1  cm.  conductor  with  a  velocity  of  1  cm.  per 
sec.,  and  moving  in  a  field  of  a  density  of  1  line  per  sq.  cm., 
the  total  cutting  of  lines  was  at  the  rate  of: 

1 X 1 X 1  =  1  line  per  second. 

But  with  our  10  cm.  conductor  moving  in  a  field  of  a  density  of 
10  000  lines  per  sq.cm.,  and  with  a  velocity  of  1000  cm.  per 
sec.,  the  total  cutting  is  at  the  rate  of: 

10  X 10  000  X 1000  =  100  000  000  lines  per  second. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR   13 

In  other  words,  the  pressure,  in  volts,  at  the  terminals  of  the 
conductor  is  equal  to  the  total  number  of  lines  cut  per  second, 
multiplied  by  10  ~8.  Let  us  now,  instead  of  considering  a  straight 
conductor,  take  a  case  where  the  conductor  is  bent  to  constitute 
the  sides  of  a  turn.  Let  this  turn  occupy  a  plane  normal  to  a 
uniform  flux.  If  in  this  position,  the  portion  of  the  flux  passing 
through  this  turn  is  1000000  lines  (or  1  megaline),  then  if  in 
1  second  the  coil  is  turned  so  that  its  plane  is  in  line  with  the 
direction  of  the  flux,  there  will  in  this  latter  position  be  no  flux 
linked  with  the  turn.  The  flux  linked  with  the  turn  will,  in  1 
second,  have  decreased  from  1  megaline  to  0.  The  conductor 
constituting  the  sides  of  this  turn  will  have  cut  1000000  lines. 
If,  during  this  1  second,  the  rate  of  cutting  is  uniform,  then  the 
pressure  at  the  terminals  will  be 

1000000XlO-8  =  0.0100  volt. 

Thus  1  turn  revolved  in  1  second  through  90  degrees  from  a 
position  where  it  is  linked  with  1  megaline  into  a  position  in 
which  it  is  parallel  to  the  flux,  will  have  induced  in  it  a  pressure  of 
0.0100  volt.  Obviously  at  this  speed,  4  seconds  are  required 
for  this  turn  to  make  one  complete  revolution,  i.e.,  to  complete 
one  cycle.  The  periodicity  is  thus  only  0.25  cycle  per  second. 
If  the  speed  is  quadrupled,  corresponding  to  one  cycle  per  second, 
then  the  pressure  will  be 

4X0.01=0.0400  volt. 

The  Pressure  Formula.  If  instead  of  having  one  turn,  we  have 
a  coil  of  T  turns,  and  if  this  coil  revolves  at  a  periodicity  of  ~ 
cycles  per  secondhand  if,  instead  of  being  linked  with  1  megaline, 
the  flux  is  M  megalines,  then  the  pressure  at  the  terminals  of  the 
winding  is  : 


This  is  the  general  formula  for  the  average  e.m.f.  induced 
in  the  windings  of  an  alternator.  But  usually  we  are  concerned 
with  the  mean  effective  value  and  not  with  the  average  value. 


14         POLYPHASE  GENERATORS  AND  MOTORS 

For  a  sine-wave  curve  of  e.m.f.  the  mean  effective  value  is  11 
per  cent  greater  than  the  average  value.  Consequently  for  the 
mean  effective  value  of  the  e.m.f.,  the  formula  becomes 

F  =  1.11X0.0400XTX~XM, 


or 


XM. 


Modification  of  Pressure  Formula.  The  formula  only  holds 
when  all  the  turns  of  the  winding  are  (when  at  the  position  of 
maximum  linkage  of  flux  and  turns),  simultaneously  linked  with 
the  total  flux  M.  This  is  substantially  the  case  with  the  windings 
of  any  one  phase  of  a  three-phase  machine.  A  reference  to  Fig. 
2  in  which  are  drawn  the  windings  of  only  one  of  the  three  phases, 
will  show  that  the  windings  of  one  phase  occupy  only  one-third 
of  the  polar  pitch  and  so  long  as  the  pole  arc  of  the  field  pole  does 
not  exceed  two-thirds  of  the  polar  pitch,  there  is  complete  linkage 
of  all  the  turns  with  the  entire  flux.  But  in  a  two-phase  machine 
(often  termed  a  quarter-phase  machine),  the  windings  of  each 
phase  occupy  one-half  of  the  polar  pitch  and  in  such  a  case  as 
that  shown  in  Fig.  3,  in  which  are  drawn  the  windings  of  only 
one  of  the  two  phases,  it  is  seen  that  the  innermost  turns  of  one 
phase  are  not  linked  with  quite  the  entire  flux. 

The  Spread  of  the  Winding.  We  express  these  facts  by  stating 
that  in  a  three-phase  winding,  the  spread  of  the  winding  is  33.3 
per  cent  of  the  polar  pitch  and  that  in  a  two-phase  winding  the 
spread  of  the  winding  is  50  per  cent  of  the  polar  pitch.  In  view 
of  the  allowances  which  must  sometimes  be  made  for  incomplete 
simultaneous  linkage  of  flux  and  turns,  it  is  preferable  to  replace 
the  formula: 


by  the  formula: 

V=KXTX~  XM. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       15 


n 


\j 


n 


\J 


3 


16 


POLYPHASE  GENERATORS  AND  MOTORS 


The  coefficient  K  for  various  winding  spreads  and  for  various 
ratios  of  pole  arc  to  pitch  may  be  obtained  by  reference  to  the 
curves  in  Fig.  4. 


10        20        30         40        50        60        70         80        90      100 
Spread  of  CoiLin  Percent  of  Pitch 

FIG.  4. — Curves  for  Obtaining  Values  of  K  in  the  Formula  V  =  KXTX~XM. 


Full  and  Fractional  Pitch  Windings.  In  the  above  statements, 
it  has  been  tacitly  assumed  that  the  windings  are  of  the  full- 
pitch  type.  In  a  full-pitch  winding,  the  center  of  the  left-hand 
side  of  a  group  of  armature  coils  is  distant  by  T,  the  polar  pitch 
from  the  center  of  the  right-hand  side  of  the  group  of  armature 
coils.  Thus  in  Fig.  5,  is  shown  a  full-pitch  winding.  But  it  is 
sometimes  desirable  to  employ  windings  of  lesser  pitch.  These 
are  called  fractional  pitch  windings.  In  Figs.  6  to  11,  are  shown 
windings  in  which  the  winding  pitches  range  from  91.5  per  cent 
down  to  50  per  cent  of  the  polar  pitch.  For  such  windings  it 
is  necessary  to  introduce  another  factor  into  the  pressure  formula 
and  we  can  employ  the  factor  K'  which  we  may  term  the  wind- 
ing-pitch factor.  If  the  winding  pitch  is  x  per  cent  then  K' 
will  be  equal  to 


sn 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       17 

MEliJ^ 

FIG.  5.  —  Diagrammatic  Representation  of  a  Full-pitch,  Three-phase  Winding. 


FIG.  6.  —  Diagrammatic  Representation  of  a  Three-phase  Winding  in  which 
the  Winding  Pitch  is  91.5  per  cent  of  the  Polar  Pitch. 


FIG.  7.  —  Diagrammatic  itepresentation  of  a  Three-phase  Winding  in  which 
the  Winding  Pitch  is  83.5  per  cent  of  the  Polar  Pitch. 


M  H  M  M  H  M  \B\  \B\  M  W  \c\  \c\  M  \c\  \A 

FIG.  8. — Diagrammatic  Representation  of  a  Three-phase  Winding  in  which 
the  Winding  Pitch  is  75  per  cent  of  the  Polar  Pitch. 


°  A  A  H  r  r  r  r  r 

FIG.  9. — Diagrammatic  Representation  of  a  Three-phase  Winding  in  which 
the  Winding  Pitch  is  66.7  per  cent  of  the  Polar  Pitch. 


C\    \0\    \A\     A     Lt     \A\    \B\    IB]    Lc     \B\    \C\    \C 


FIG.  10. — Diagrammatic  Representation  of  a  Three-phase  Winding  in  which 
the  Winding  Pitch  is  58.5  per  cent  of  the  Polar  Pitch. 


r      r 


FIG.  11.  —  Diagrammatic  Representation  of  a  Three-phase  Winding  in  which 
the  Winding  Pitch  is  50.0  per  cent  of  the  Polar  Pitch. 


18         POLYPHASE  GENERATORS  AND  MOTORS 

Thus  when  the  winding  pitch  is  80  per  cent,  the  winding  -pitch 
factor  is  equal  to 

sin          X90°    =  sin  72°  =  0.951. 


Digression  Regarding  Types  of  Windings.  An  examination  of 
Fig.  5  will  show  that  the  conductors  contained  in  any  one  slot 
belong  exclusively  to  some  one  of  the  three  phases.  On  the  contrary 
in  the  case  of  fractional-pitch  windings,  there  are  a  certain  propor- 
tion of  the  slots  containing  conductors  of  two  different  phases. 
As  a  consequence,  fractional-pitch  windings  should  be  of  the  two- 
layer  type.  This  leads  to  the  use  of  "  lap  "  windings  such  as  are 
employed  for  continuous-electricity  machines.  But  full-pitch  wind- 
ings may  be  carried  out  either  as  "  lap  "  windings  or  as  "  spiral  " 
windings,  and  consequently  as  single-layer  or  as  two-layer  windings. 

The  two  varieties  are  shown  diagrammatically  in  Figs.  12 
and  13  on  the  following  page. 

While  dealing  with  this  subject  of  windings  the  occasion  is 
appropriate  for  explaining  another  terminological  distinction. 
Windings  may  be  carried  out  in  the  manner  termed  "  whole- 
coiled,"  in  which,  for  any  one  phase,  there  is  an  armature  coil 
opposite  each  field  pole,  as  indicated  in  Fig.  14,  or  they  may 
be  carried  out  as  half-coiled  windings,  there  being,  for  any  one 
phase,  only  one  armature  coil  per  pair  of  poles  instead  of  one 
armature  coil  per  pole.  A  half-coiled  winding  is  shown  in  Fig. 
15.  It  is  usual  to  employ  half-coiled  windings  for  three-phase 
generators  where  the  spiral  type  is  adopted.  But  lap  windings 
are  inherently  whole-coiled  windings  as  may  be  seen  from  an 
inspection  of  Fig.  12. 

It  is  mechanically  desirable  to  employ  fractional-pitch  windings 
in  bipolar  designs,  as  otherwise  the  arrangement  of  the  end 
connections  presents  difficulties.  In  most  instances  of  designs  for 
more  than  two  poles,  a  full-pitch  winding  is  quite  as  suitable  as 
any  other.  Let  us  employ  a  full-pitch  winding  in  our  2500-kva. 
machine. 

Estimation  of  the  Flux  per  Pole.  We  can  now  estimate  the 
flux  required,  when,  at  no  load,  the  terminal  pressure  is  12  000 
volts.  This  pressure  corresponds  to 

12000 
—  y=r-  =  6950  volts  per  phase. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR        19 


FIG.  12.— 6-Pole  Lap  Winding  with  72  Conductors. 


B 


FIG.  13.— 6-Pole  Spiral  Winding  with  72  Conductors. 


20         POLYPHASE  GENERATORS  AND  MOTORS 
Therefore 

F  =  6950. 

We  have  determined  upon  employing  10  conductors  per  slot 
and  5  slots  per  pole  per  phase.  Consequently  for  T,  the  number 
of  turns  in  series  per  phase,  we  have,  (since  the  machine  has  8 
poles) : 


FIG.  14.— Whole-coiled  Spiral 
Winding. 


FIG.  15.— Half-coiled  Spiral 
Winding. 


Thus  we  have: 


Therefore 


6950  =  0.0444  X  200  X  25  XM. 


M  = 


6950 


0.0444X200X25 
31.3  megalines. 


31.3  megalines  must,  at  no  load,  enter  the  armature  from  each 
field  pole. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       21 

The  I R  Drop  at  Full  Load.  At  full  load  we  must  make  an 
allowance  for  the  IR  drop  in  the  armature  winding.  We  have 
previously  ascertained  that  the  full-load  current  is 

I  =  120  amperes. 

Estimation  of  the  Mean  Length  of  Turn  and  of  the  Armature 
Resistance.  We  must  now  estimate  the  armature  resistance. 
A  convenient  empirical  formula  for  estimating  the  mean  length 
of  one  turn  (mlt)  is: 


Appropriate  values  for  K  for  machines  for  various  pressures  are 
given  in  Table  6: 

TABLE  6. — VALUES  OF  K  IN  FORMULA  FOR  MLT. 


Terminal  Pressure  for 
which  the  Stator  is 
Wound. 

Value  of  K  in  the  Formula: 
mlt.  =2\o  +  Kr. 

500 

2.5 

1000 

3.0 

2000 

.3.5 

4000 

4.0 

6000 

4.5 

8000 

5.0 

10000 

5.5 

12000 

6.0 

Thus  for  our  2500-kva.  design  we  have : 

mlt.  =2X118+6.0X70 
=  236+420 
=  656  cm. 

The  cross-section  of  the  conductor  is  (1.2X0. 33  =  )0. 396  sq.  cm. 

It  is  a  good  plan  from  the  standpoint  of  uniformity,  to    figure 

out  all  warm  resistances  (of  the  usual  run  of  dynamo-electric 


22         POLYPHASE  GENERATORS  AND  MOTORS 

machinery),  to  correspond  to  a  temperature  of  60°  Cent.  At  this 
temperature,  the  specific  resistance  of  commercial  copper  is : 

0.00000200  ohm  per  crn.  cube. 
At  60°  Cent,  the  resistance  per  phase,  is,  for  our  2500-kva.  machine : 

0.00000200X656X200 
Resis.  =  -  ~~o~3Qfi~ 

=  0.665  ohm. 
Consequently,  at  full  load,  the  IR  drop  per  phase  is : 

120X0.665  =  80.0  volts. 
Thus  at  full  load  the  internal  pressure  is: 

6950+80  =  7030  volts  per  phase. 
At  full  load  then,  the  flux  entering  the  armature  from  each  pole  is : 

7030 
"0.0444X200X25 

=  31.6  megalines. 

In  this  particular  case,  the  internal  drop  is  so  small  that  it  is 
hardly  worth  while  to  distinguish  between  the  full-load  and  no- 
load  values  of  the  flux  per  pole.  But  often  (particularly  in  small, 
slow-speed  machines),  there  is  a  more  appreciable  difference 
which  must  be  taken  into  account. 

The  Design  of  the  Magnetic  Circuit.  We  are  now  ready  to 
undertake  the  design  of  the  magnetic  circuit  for  our  machine. 
In  the  stator,  the  magnetic  circuit  must  transmit  a  flux  of  31.6 
megalines  per  pole.  This  is  the  flux  which  must  become  linked 
with  the  turns  of  the  armature  winding.  But  a  somewhat  greater 
flux  must  be  set  up  in  the  magnet  core,  for  on  its  way  to  the  arma- 
ture this  flux  experiences  some  loss  through  magnetic  leakage. 
The  ratio  of  the  flux  originally  set  up  in  the  magnet  core  to  that 
finally  entering  the  armature  is  termed  the  leakage  factor. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR        23 

Leakage  Factor.  In  our  machine,  we  shall  assume  the  leakage 
factor  to  be  1.15;  that  is  to  say,  the  quantity  of  flux  taking  its 
rise  in  the  magnet  core  is  15  per  cent  greater  than  the  portion 
finally  entering  the  armature  and  becoming  linked  with  the  turns 
of  the  armature  windings. 

Consequently  the  cross-sections  of  the  magnet  cores  and  of 
the  yoke  must  be  proportioned  for  transmitting  this  15  per  cent 
greater  flux. 

1.15X31.6  =  36.3  megalines. 

Material  and  Shape  of  Magnet  Core.  We  shall,  in  this 
instance,  make  the  magnet  core  of  cast  steel  and  we  shall  employ 
a  magnetic  density  of  17  500  lines  per  sq.cm.  Consequently  we 
shall  require  to  provide  a  cross-section  of 

36  300  000 

sq.cm. 


A  circular  magnet  core  of  this  cross-section,  would  have  a 
diameter  of 


1X208U     ri    . 

—  =  51.4  cm. 


At  the  air-gap,  the  polar  pitch,  T,  is  equal  to  70  cm.,  but  it 
becomes  smaller  as  we  approach  the  inner  ends  of  the  magnet 
cores  and  there  would  not,  in  our  machine,  be  room  for  cores  of 
a  diameter  of  51.4  cm.,  even  aside  from  the  space  required  on  these 
cores  for  the  magnet  windings.  Let  us  take  as  a  preliminary 
assumption  for  the  radial  length  of  the  magnet  cores  (including 
pole  shoes)  28  cm.  This  gives  a  diameter  at  the  inner  ends  of 
the  magnet  cores  (neglecting  the  radial  depth  of  the  air-gap), 
of 

178- 2X28  =  122  cm.  * 
The  polar  pitch  at  this  diameter  is : 


24 


POLYPHASE  GENERATORS  AND  MOTORS 


Obviously,  then,  we  could  not  employ  a  magnet  core  of  circular 
cross-section,  for  we  have  seen  that  its  diameter  would  require  to 
be  51. 4  cm.  Out  of  the  total  available  circumferential  dimen- 
sion of  48.0  cm.,  let  us  take  the  lateral  dimension  of  the  magnet 
core  as  26  cm.,  and  let  us  constitute  the  section  of  the  magnet 
core,  of  a  rectangle  with  a  semi-circle  at 
each  end,  as  shown  in  Fig.  16.  The  diam- 
eter of  the  semi-circle  is  26  cm.  Conse- 
quently the  area  provided  by  the  two 
semi-circles  amounts  to: 


-26  cm- 


FIG.  16. — Cross-section 


=  530sq.cm. 

This  leaves 

2080 -530  =  1550  sq.cm. 

to  be  provided  by  the  rectangle.     Thus  the 
length  of  the  rectangle  (parallel  to  the  shaft) 


of  Magnet  Core  of     must  be: 
8-pole,    375    r.p.m., 
2500-kva.        Three- 
phase  Generator. 


1550 
26 


=  59.6  cm.  (or  60  cm.). 


The  overall  length  of  the  magnet  core  is,  then,  26+60  =  86  cm. 
The  pole  shoe  may  be  made  114  cm.  long,  thus  projecting: 


114-86 


=  14  cm. 


at  each  end  of  the  magnet  core.  Let  us  make  the  pole  arc  equal 
to  42  cm.  This  is  |^X100  =  J  60  per  cent  of  the  pitch  at  the 

air  gap.  We  may  allow  8  cm.  for  the  radial  depth  of  the  pole 
shoe  at  the  center.  This  leaves  28  —  8  =  20  cm.  for  the  radial 
length  along  the  magnet  core  which  is  available  for  the  magnet 
winding.  The  depth  available  for  the  winding  is — at  the  lower 
end  of  the  magnet  core: 

48-26 

-  =11  cm. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       25 


If  we  make  our  magnet  spool  winding  of  equal  depth  from  top 
to  bottom,  then,  allowing  a  centimeter  of  free  space  at  the  lower 
end  of  the  spool,  the  two  dimensions  of  the  cross-section  of  the 
spool  winding  will  be  : 

20  cm.XlO  cm. 

It  remains  to  be  ascertained  at  a  later  stage  whether  this  space 
is  sufficient  to  accomodate  the  required  ampere-turns. 

The  Cross-section  of  the  Magnetic  Circuit  at  the  Stator  Teeth. 
The  next  part  of  the  magnetic  circuit  which  we  should  investigate, 
is  the  section  at  the  stator  teeth.  We  must  first  determine  upon 
suitable  proportions  forthe  ventilating  ducts.  A  suitable  number 
of  ventilating  ducts  may  be  arrived  at  from  the  data  in  Table  7. 

TABLE  7.  —  DATA  REGARDING  VENTILATING  DUCTS. 


Peripheral 
Speed  in  Meters 
per  Second. 

Number  of  Ventilating  Ducts,  Each  15  mm.  Wide,  per  Decimeter  of 
Gross  Core  Length. 

For  \g  =20. 

ForX0=50. 

ForX0=80. 

For  X(7=150. 

10 

2.1 

2.3 

2.5 

2.7 

20 

1.5 

1.8 

2.1 

2.3 

30 

1.1 

1.4 

1.7 

1.9 

40 

0.8 

1.1 

1.4 

1.6 

60 

0.6 

0.9 

1.2 

1.4 

80 

0.4 

0.7 

1.0 

1.2 

In  order  to  apply  the  data  in  this  table,  we  require  to  know 
the  peripheral  speed. 

Determination  of  the  Peripheral  Speed.  Not  having  yet 
determined  upon  the  radial  depth  of  the  air-gap,  let  us  still 
neglect  it  and  estimate  the  peripheral  speed  from  the  internal 
diameter  of  the  stator.  We  have  : 

D  =  178  cm. 

375 
Peripheral  speed  =  1.78XxX-^r  =  35  meters  per  sec. 


Let  us,  then,  arrange  for  1.7  ventilating  ducts  per  decimeter  of 
gross  core  length. 


11.8X1.7  =  20. 


26         POLYPHASE  GENERATORS  AND  MOTORS 

We  shall  provide  20  ducts  and  each  duct  shall  have  a  width  of 
15  mm.  Thus  the  aggregate  width  occupied  by  ventilating 
ducts  is  20X1.5  =  30  cm.  But  10  per  cent  of  the  "  apparent  " 
thickness  of  the  core  laminations  is  occupied  by  layers  of  varnish 
by  means  of  which  the  sheets  are  insulated  from  one  another  in 
order  to  prevent  eddy  currents. 

The  Net  Core  Length.  The  length  of  active  magnetic  material, 
after  deducting  the  space  occupied  by  the  ventilating  ducts  and 
by  the  insulating  varnish,  may  be  termed  the  net  core  length  and 
may  be  designated  by  \n.  For  our  2500-kva  machine  we  have  : 

<rt  =  ;113-30)X0.90  =  79.2  cm. 

The  width  of  the  stator  slot  has  been  calculated  and  has  been 
ascertained  to  be  24.1  mm.  There  are  15  teeth  per  pole  (since  there 
are  5  slots  per  pole  per  phase).  The  tooth  pitch  at  the  air-gap  is 


.     mm. 
lo 

Consequently  the  width  of  each  tooth  is 

(46.6-24.1  =)22.5  mm. 
The  aggregate  width  of  the  15  teeth  is 

15X2.25  =  33.8  cm. 

Thus  the  gross  cross-section  per  pole,  at  the  narrowest  part 
of  the  stator  teeth,  is 

79.2X33.8  =  2680  sq.cm. 

But  only  a  portion  of  this  section  will  be  employed  at  any  one 
time  for  transmitting  the  flux  per  pole,  for  the  pole  arc  is  only 
60  per  cent  of  T.  The  portion  directly  opposite  the  pole  face  is 

0.60X2680  =  1610  sq.cm. 

On  the  other  hand,  the  lines  will  spread  considerably  in  crossing  the 
gap,  and  this  spreading  will  increase  the  cross-section  of  the  stator 
teeth  utilized  at  any  instant  by  the  flux  per  pole.  Let  us  in 
our  machine  take  the  spreading  factor  equal  to  1.15.  Conse- 
quently we  have  : 

Cross-section  of  magnetic  circuit  at  stator  teeth  = 
1.15X1610  =  1850  sq.cm. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR        27 


The  density  at  the  narrowest  part  of  the  stator  teeth  is: 
31  600  000 


1850 


=  17  100  lines  per  sq.cm. 


This  is  a  suitable  value;   indeed  a  slightly  higher  value,  say  any- 
thing up  to  19  000  lines  per  sq.cm.,  could  have  been  employed. 

Density  and  Section  in  Stator  Core.  For  the  density  in  the 
stator  core,  back  of  the  teeth,  we  may  take  10  000  lines  per  sq.cm. 
The  total  flux  in  the  stator  is  31.6  megalines  per  pole.  This  flux, 
after  passing  along  the  teeth,  divides  into  two  equal  parts  and 
flows  off  to  right  and  left  to  the  adjacent  poles  on  either  side, 
as  indicated  diagrammatically  in  Fig.  17.  Consequently  the 
magnetic  cross-section  required 
in  the  stator  core  is: 


31  600  OOP 
2X10000 


=  1580  sq.cm. 


Since  \n  is  equal  to  79.2  cm. 
the  radial  depth  of  the  stator 
core  back  of  the  slots  must  be 

!lT20cm- 


FIG.  17. — Diagrammatic  Representation 
of  the  Path  of  the  Magnetic  Flux  in 
the  Magnet  Core,  Pole  Shoe,  Air-gap, 
Stator  Teeth,  and  Stator  Core. 


The   slot   depth   is   5.72    cm. 

Consequently  the  total  radial 

depth  of  the  stator  core  from  the  air-gap  to  the  external  periphery 

is  20.0+5.7  =  25.7  cm.     D,    the  internal  diameter  of  the  stator 

punching,   is   178  cm.     Consequently  the  external  diameter  of 

the  stator  punching  is: 

178+2X25.7  =  229.4  cm.  (or  230  cm.) 

In  Fig.  18  is  given  a  drawing  of  the  stator  punching,  and  in  Fig. 
19  is  shown  a  section  through  the  stator  core  with  its  20  ventilating 
ducts. 

Even  now  we  are  not  quite  ready  to  consider  the  question  of 
assigning  a  suitable  value  to  the  depth  of  the  air-gap.  But 
pending  arriving  at  the  right  stage  of  the  calculation,  let  us  pro- 
ceed on  the  basis  that  the  depth  of  the  air-gap  is '2  cm.  This 
permits  us  to  list  a  number  of  diametrical  measurements,  as 
follows : 


28         POLYPHASE  GENERATORS  AND  MOTORS 

Preliminary  Tabulation  of  Leading  Diameters. 

External  diameter  of  stator  core 2300  mm. 

Diameter  at  bottom  of  stator  slots  (1780+2x57  =  ). .  .  1894  mm. 

Internal  diameter  of  stator  (D) 1780  mm. 

External  diameter  of  rotor  (1780-2X20) 1740  mm. 

Diameter  to  bottom  of  pole  shoes  (1740-2X80) 1580  mm. 

Diameter  to  bottom  of  magnet  cores  (1580  —  2X200). .  1180  mm. 


FIG.  18. — Dimensions  of  Stator  Lamination  for  8-pole,  375-r.p.m.,  2500-kva., 
Three-phase  Generator. 


-15  mm 


FIG.  19. — Section  through  Stator  Core  of  8-pole,  375-r.p.m.,  2500-kva.,  Three 
phase  Generator,  showing  the  20  Ventilating  Ducts. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       29 

An  end  view  of  the  design  of  the  magnetic  circuit  so  far  as  we 
have  yet  proceeded,  is  shown  in  Fig.  20. 

We  are  now  ready  to  work  out  the  cross-section  of  the  so- 
called  "  magnet  yoke  "  which  completes  the  magnetic  circuit 
between  the  inner  extremities  of  the  magnet  cores.  We  may 
employ  a  density  of  12  000  lines  per  sq.cm.  Here  also  the  flux 
from  any  one  magnet  core  divides  into  two  equal  halves  which 
flow  respectively  to  the  right  and  to  the  left  on  their  wav  to  the 


FIG.  20  —  End  View  of  Design  of  Magnetic  Circuit  of  8-poie,  375-r.p.m.,  250G 
kva.,  Three-phase  Alternator. 

adjacent  magnet   cores  on  either   side.      Consequently    for  the 
magnet  yoke,  we  require  a  cross-section  of 

36  300  000 


For  the  dimension  parallel  to  the  shaft,  let  us  employ  120  cm. 
The  radial  depth  of  the  magnet  yoke  should  consequently  be: 


1510 
120 


12.6  cm. 


Since  the  external  diameter  of  the  magnet  yoke  is  1180  mm., 
the  internal  diameter  is  1180-2X126  =  928  mm.     The  complete 


30         POLYPHASE  GENERATORS  AND  MOTORS 


CALCULATIONS  FOR  2500-KVA.  GENERATOR        31 

list  of  leading  diameters  of  the  parts  of  the  magnet  circuit,  is, 
(pending  arriving  later  at  a  final  value  for  the  air-gap  depth), 
as  follows : 

Extended  Preliminary  Tabulation  of  Leading  Diameters. 

External  diameter  of  stator  core 2300  mm. 

Diameter  at  bottom  of  stator  slots 1894  mm. 

Internal  diameter  of  stator  (D) 1780  mm. 

External  diameter  of  rotor 1740  mm. 

Diameter  to  bottom  of  pole  shoes 1580  mm. 

External  diameter  of  magnet  yoke 1180  mm. 

Internal  diameter  of  magnet  yoke 928  mm. 

These  diameters  and  the  leading  dimensions  parallel  to  the 
shaft,  are  indicated  in  the  views  in  Fig.  21. 

Mean  Length  of  Magnetic  Circuit.  The  next  step  is  to  ascer- 
tain the  lengths  of  the  various  parts  of  the  magnetic  circuit,  i.e., 
of  the  mean  path  followed  by  the  magnetic  lines.  This  mean 
path  is  indicated  in  Fig.  22.  The  mmf.  in  the  field  winding  on 


FIG.  22. — Diagram  showing  the  Mean  Path  Followed  by  the  Magnetic  Lines 
in  an  8-pole  Generator  with  an  Internal  Revolving  Field. 

any  one  of  the  magnet  poles  has  the  task  of  dealing  with  just 
one-half  of  the  complete  magnetic  circuit  formed  by  two  adjacent 
poles.  The  lengths  which  we  desire  to  ascertain  are  the  lengths 
corresponding  to  such  a  half  circuit  as  indicated  by  the  heavy 
lines  in  Fig.  22. 


32 


POLYPHASE  GENERATORS  AND  MOTORS 


It  is  amply  exact  for  our  purpose  to  take  the  lengths  in  the 
stator  core  and  in  the  magnet  yoke  as  equal  to  the  mean  circum- 
ferences in  those  parts,  divided  by  twice  the  number  of  poles. 
The  mean  diameters  in  these  parts  are; 

230.0+189.4 
Mean  diameter  in  stator  core  — •= —    -  =209.7  cm. 

118.0+92.8 
magnet  yoke    =  —  -  =  105.4  cm. 


The  machine  has  8  poles.     Consequently: 

Mean  length  magnetic  circuit  in  stator  core  =  —      7—  =41  cm. 


Mean  length  magnetic  circuit  in  magnet  yoke  =  —  '       ''  =  21  cm. 


We  also  have: 

Mean  length  of  magnetic  circuit  in  teeth  =5.7  cm. 

Mean  length  of  magnetic  circuit  in  magnet  core  =  20  cm. 

The  pole  shoe  is  so  unimportant  a  part  that  we  may  in  the  cal- 
culation of  the  required   magnetomotive    forces  (mmf.)  neglect 
it.     As  to  the  air-gap,  we  must,  for  reasons  which  will  be  under- 
stood later,  still  defer  taking  up  the  calculations  relating  to  it. 
We  may  now  make  up  the  following  Table  : 


TABULATION  OF  DATA  FOR  MMF.  CALCULATIONS. 


Designation  of  Parts 
of  Magnetic  Circuit. 

Lengths  in  Centi- 
meters of  Parts  of 
Magnetic  Circuit. 

Cross-sections,  in 
Square  Centimeters 
of  Parts  of  Mag- 
netic Circuit. 

Densities  in  Lines  per 
Square  Centimeter 
of  Parts  of  Mag- 
netic Circuit. 

Teeth  

5.7 

1850 

17  100 

Magnet  core  
Magnet  yoke.  .  .  . 
Stator  core 

20 
21 
41 

2080 
1510 
1580 

17500 
12000 
10000 

Saturation  Data  of  Magnetic  Materials.     In  order  to  ascer- 
tain the  mmf.  required  in  each  of  these  portions  of  the  magnetic 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       33 

circuit,  we  must  consult  saturation  data  of  the  materials  of  which 
these  parts  are  built.     Appropriate  values  are  given  in  Table  8: 


TABLE  8.     MAGNETOMOTIVE  FORCE  PER  CM.  FOR  VARIOUS  MATERIALS. 


Density  in  Lines  per 
Square  Centimeter. 

Magnetomotive  Force  in  Ampere-turns  per  Centimeter 
of  Length. 

Forgings  and  Sheets 
of  Wrought  Iron 
or  Steel. 

Cast  Steel. 

Cast  Iron. 

1000 
2000 
3000 

0.6 
1.0 

2 
3 
4 

3 
7 
11 

4000 
5000 
6000 

1.5 
1.9 
2.3 

5 
6 

7 

17 
24 
33 

7000 
8000 
9000 

2.7 
3.0 
3.6 

7 
8 
9 

46 
62 
90 

10000 
11000 
12000 

4.7 

5.8 
7.0 

10 
12 
15 

150 

13000 
14  000 
15000 

9.0 
12 
17 

20 

27 
37 

16000 
17000 
18000 

25 
55 
95 

50 
80 
130 

19000 
20000 
21000 

190 
300 
470 

22000 
23000 
24000 

670 
980 
1500 

Estimation  of  Component  mmf.  With  these  data  we 
can  estimate  the  mmf.  required  for  each  part  of  the  magnetic 
circuit.  Thus  since  the  tooth  density  is  17  100  and  since  the 
core  is  built  up  of  sheet  steel,  there  will  be  required  a  mmf. 
of  57  ampere-turns  (ats.)  per  cm.  of  length.  The  length  of 


34 


POLYPHASE  GENERATORS  AND  MOTORS 


the  tooth  is  5.7  cm.,  consequently  the  total  mmf.  required  for 
the  teeth  is 

5.7X57  =  325  ats. 

Similar  calculations  may  be  made  for  the  other  parts.  It  is  con- 
venient to  arrange  these  calculations  in  some  such  tabular  form  as 
the  following: 


Designation  of  the  Part  of  the 
Magnetic  Circuit. 

Density  in 
Lines  per 
Square 
Centimeter. 

Mmf.  in 
Ampere- 
turns  per 
Centimeter. 

Length  in 
Centi- 
meters. 

Required 
Mmf. 

Teeth 

17  100 

57 

5   7 

330  ats 

Magnet  core  
Magnet  yoke 

17500 
12000 

100 
15 

20 
21 

2000   " 
320   " 

Stator  core                           .  .  . 

10000 

4.7 

41 

190  " 

Total  mmf  for  these  four  parts  = 

2840  ats 

The  mmf.  of  Armature  Interference.  There  is  another  matter 
to  be  dealt  with  before  we  can  proceed  with  the  air-gap  calcula- 
tions. This  relates  to  the  mmf.  of  the  armature  winding  and  is 
of  importance.  At  load  it  interferes  with  the  operation  of  the 
mmf.  emanating  from  the  field  spools.  We  have  arranged  for  a 
winding  with  120  slots  and  10  conductors  per  slot.  Thus  we  have 
a  total  of  10  X 120  =  1200  armature  conductors,  or 


1200 


=  600  turns. 


This  comes  to: 


and 


--  =  200  turns  per  phase 
o 


200 

—^—  =  25  turns  per  pole  per  phase. 


/,  the  full-load  current,  is  equal  to  120  amperes.  Therefore  we 
have  25X120  =  3000  ats.  per  pole  per  phase.  This  is  termed 
the  armature  strength.  Many  designers  resort  to  theoretical 
reasoning  in  ascertaining  from  the  mmf.  of  one  phase  of  the  arma- 
ture winding,  the  resultant  mmf.  exerted  by  the  three  phases. 
But  in  practice,  the;  distribution  of  the  stator  and  rotor  windings, 
the  ratio  of  the  pole  arc  to  the  pitch,  and  other  details  of  the  design, 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       35 

so  complicate  the  matter  as  to  considerably  invalidate  any  theo- 
retical deductions.  But  working  backward  from  experimental 
observations  it  has  been  ascertained  that,  independently  of  the 
various  relative  dispositions  of  the  windings  and  of  other  factors 
of  the  design,  results  consistent  with  practice  may  be  arrived  at 
by  taking  as  the  resultant  mmf .  of  the  three  phases,  2.4  times  the 
mmf.  of  each  phase.  Consequently,  for  our  design  we  have, 
at  full  load,  an  armature  mmf.  of 

2.4X3000  =  7200  ats. 

It  is  only  at  zero  power-factor  that  these  armature  ats.  have  the 
same  axis  as  the  field  ampere-turns.  If,  when  the  power-factor  of 
the  external  load  is  zero,  the  output  is  120  amperes,  then,  if  the  cur- 
rent is  lagging,  the  resultant  mmf.  acting  to  send  flux  round  the 
magnetic  circuit  is  obtained  by  subtracting  7200  ats.  from  the 
excitation  on  each  field  pole.  If  each  field  coil  provided  a  mmf. 
of  15  000  ats.,  then  the  resultant  mmf.  would,  for  this  lagging 
lead  of  120  amperes,  be 

15  000 -7200  =  7800  ats. 

If  the  120  amperes  were  leading  and  if  the  power-factor  were  zero, 
then  the  resultant  mmf.  would  be 

15  000+7200  =  22  200  ats. 

For  120  amperes  at  other  than  zero  power-factor,  the  armature 
mmf.  does  not  affect  the  resultant  mmf.  to  so  great  an  extent. 
In  a  later  section,  we  shall  deal  with  a  method  of  determining  the 
extent  of  the  influence  of  the  armature  mmf.  when  the  power- 
factor  is  other  than  zero. 

Even  at  this  stage  it  is  very  evident  from  the  phenomena  that 
have  been  considered,  that  the  armature  mmf.  will  at  heavy  loads, 
and  especially  at  overloads,  exert  a  less  disturbing  influence  the 
greater  the  mmf.  provided  on  the  field  spools,  and  that  for  a  given 
all-around  quality  of  pressure  regulation,  the  higher  the  armature 
mmf.,  the  higher  should  be  the  field  mmf.  A  rule  usually  leading 
to  suitable  pressure  regulation,  at  usual  power-factors,  is  to  employ 
in  each  field  spool  a  mmf.  equal  to  twice  the  armature  strength. 
A  lower  ratio  is  often  employed;  indeed  it  is  often  impracticable 
to  find  room  for  field  spools  supplying  so  high  a  mmf.  But  let 


36         POLYPHASE  GENERATORS  AND  MOTORS 

us  endeavor  to  adhere  to  this  ratio  in  the  case  of  our  example. 
Thus  our  field  mmf .  should  be 

2X7200  =  14400ats. 

But  we  have  seen  that  the  iron  parts  of  our  magnetic  circuit  only 
require  a  total  mmf.  of  2840  ats.  How  then  can  we  employ  a 
total  mmf.  of  14  400  ats.  and  not  obtain  through  the  armature 
winding,  a  greater  flux  than  the  31.3  megalines  which  we  have 
found  to  correspond  to  the  required  pressure  of  12  000  volts 
(6950  volts  per  phase)? 

We  can  so  design  the  air-gap  as  regards  density  and  length,  as 
to  use  up  the  remaining  14  400  —  2840  =  11  560  ats.,  in  overcoming 
its  magnetic  reluctance. 

The  Estimation  of  the  Air-gap  Density.  First  let  us  estimate 
the  air-gap  density.  We  have 

T  =  70cm. 

Pole  arc  =  0.60X70  =  42  cm. 

Length  of  the  pole  shoe  parallel  to  the  shaft  =  114  cm. 
Area  of   pole   face  =  42X1 14  =  4800  sq.cm. 

31300000     «Konl. 
Density  in  pole  face  =       .„     —  =  6520  lines  per  sq.cm. 

Many  designers  employ  complicated  methods  for  estimating  the 
density  in  the  air-gap.  These  methods  involve  introducing 
"  spreading  coefficients  "  to  allow  for  the  flaring  of  the  lines  after 
they  have  emerged  from  the  pole  face.  They  also  involve  cal- 
culations of  the  area  and  density  of  the  flux  where  it  enters  the 
armature  surface.  The  author's  own  custom,  in  the  case  of  alter- 
nators, is  to  usually  take  the  air-gap  density  as  equal  to  the  pole- 
face  density,  though  it  is  quite  practicable  afterward  to  use 
one's  judgment  in  taking  a  somewhat  higher  or  lower  value 
according  as  the  other  conditions  indicate  that  the  pole-face 
density  would  be  lower  or  higher  than  the  air-gap  density.  In 
this  machine,  we  need  make  no  corrections  of  this  sort,  but  may 
take  the  air-gap  density  as  6520  lines  per  sq.cm.  This  will 
require  a  mmf.  of 

^X  6520  =  5200  ats. 
per  cm.  of  radial  depth  of  the  air-gap. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR          37 

The  Radial    Depth  of   the   Air-gap.    Evidently  the  air-gap 
should  have  a  radial  depth  of: 


2.22  cm.  =22.2  mm. 


Since  such  calculations  are  necessarily  only  very  rough,  we  shall 
do  well  to  modify  this  and  make  the  air-gap  only  18  mm.  deep. 
If,  when  the  machine  is  tested,  we  find  it  desirable,  we  can 
increase  the  air-gap  by  turning  down  the  rotor  to  a  slightly 
smaller  diameter. 

The  radial  depth  of  the  air-gap,  in  mm.,  may  be  denoted  byA. 
For  our  machine 

A  =  18. 

Revised  Tabulation  of  Leading  Diameters.  Having  now 
determined  upon  a  value  for  the  radial  depth  of  the  air-gap,  let 
us  again  tabulate  the  various  diameters  in  our  machine  : 

External  diameter  of  stator  core  ....................  2300  mm. 

Diameter  at  bottom  of  stator  slots  .................  1894  mm. 

Internal  diameter  of  stator  (D)  ....................  1780  mm. 

External  diameter  of  rotor    (D  —  2A)    [Revised    from 

earlier  table]  .................................  1744  mm. 

Diameter  to  bottom  of  pole  shoes  ............  ......  1580  mm. 

External  diameter  of  magnet  yoke  ..................  1180  mm. 

Internal  diameter  of  magnet  yoke  ..................  928  mm. 

Saturation  Curves  at  No  Load.  The  next  step  is  to  work 
out  data  from  which  we  can  construct  a  no-load  saturation  curve 
for  our  machine,  i.e.,  a  curve  in  which  ordinates  will  indicate  the 
pressure  per  phase,  in  volts,  and  abscissae  will  indicate  magneto- 
motive force  per  field  spool  in  ats.  We  have  previously  obtained 
one  point  on  this  curve,  namely  the  point  for  which  we  have 
found  ordinate,  6950  volts;  abscissa,  14  400  ats.  The  process  can 
be  considerably  abbreviated  in  obtaining  further  points.  Let  us 
work  out  the  mmf.  required  for  6000  volts,  and  for  7500  volts. 
With  these  three  points  we  shall  be  able  to  construct  the  no-load 
saturation  curve. 


38         POLYPHASE  GENERATORS  AND  MOTORS 


For  the  air-gap  mmf  .  we  obtain  the  desired  results  by  simple 
proportion.     Thus 


For  6000  volts:  'Air-gap  ats.  = 


For  7500  volts:  Air-gap 


560  =  10  000  ats. 


560  =  12  500  ats. 


But  for  the  iron  parts  it  is  necessary  first  to  obtain  the  flux  density 
by  simple  proportion  and  then  from  Table  8,  on  page  33,  obtain 
the  corresponding  values  of  the  mmf.  Thus  for  the  teeth  we  have  : 

For  6950  volts:     mmf.      =57  ats.  per  cm.  (from  p.  34). 

fiOOO 
For  6000  volts  :     Density  =  ^  X  17  100  =  14  800  lines  per  sq.cm. 


Corresponding  mmf.          =16  ats.  per  cm. 

7500 


For  7500  volts :     Density 


6950 


X17  100=  18  400  lines  per  sq.cm. 


Corresponding  mmf.          =  125  ats.  per  cm. 
Since  the  length  is  5.7  cm.,  we  have: 
Total  mmf.  for  teeth: 

For  6000  volts :     16  X  5.7  =  90  ats. 
For  6950  volts :     57  X  5.7  =  330  ats. 
For  7500  volts :     125  X  5.7  =  720  ats. 

In  the  same  way  we  may  estimate  the  corresponding  values  for 
magnet  core,  magnet  yoke,  and  stator  core.  It  is  needless  to 
record  the  steps  in  these  calculations.  The  results  are  brought 
together  in  the  following  table: 


6000  Volts. 

6950  Volts. 

7500  Volts. 

Air-gap  

10  000  ats. 

11  560  ats. 

12  500  ats. 

Teeth  
Magnet  core  
Magnet  yoke  .... 
Stator  core  

90  " 
770  " 
200  " 
140  " 

330   " 
2000  " 
320  " 
190  " 

720  " 
5300  " 
^420  " 
230  " 

Total  mmf  

11  200  ats. 

14  400  ats. 

19  170  ats. 

CALCULATIONS  FOR  2500-KVA.  GENERATOR       39 

The  no-load  saturation  curve  in  Fig.  23  has  been  plotted  from 
these  results. 


8,000 
7,000 
6,000 

75  5,000 
> 

.s 

2 

|  4,000 
|  3,000 
2,000 
1,000 

*.     ~~ 

^.    — 

__        — 

_         - 

m           '• 

X 

^ 

A 

(S 

A 

' 

/ 

/ 

/ 

/ 

/ 

/ 

1 

/ 

y 

/ 

/ 

/ 

/ 

1 
c^ 

iiiiiiiiiii 

1    1   §    §   i 

I           -rjT          CD            CO           C> 
j           ^          CM           51          CO 

i    V     w    «     3     3     ;£    £    «$     £    { 

mmf  per  Field  Spool  in  ats 

FIG.  23. — No-load  Saturation  Curve  of  2500-kva.  Generator  with  an  Air-gap 

of  18  mm. 


THE    PRESSURE    REGULATION 

The  subject  of  this  section  is  one  to  which  much  study  and 
discussion  have  been  devoted.  Nevertheless  we  still  find  wide 
differences  of  opinion  with  reference  to  the  problems  involved  in 
the  estimation  of  the  excitation  required  in  any  given  case  and 
for  any  given  conditions  of  operation. 

The  method  which  will  be  described  has  the  merit  of  brevity 
combined  with  at  least  as  much  exactness,  so  far  as  regards  the 
results  obtained,  as  can  be  shown  to  be  possessed  by  any  other 
method.  The  chief  defect  in  the  method  relates  to  the  theoretical 
indefensibility  from  the  quantitative  standpoint  of  certain  steps 


40         POLYPHASE  GENERATORS  AND  MOTORS 

in  the  calculations.  Since,  however,  the  occurrences  assumed  to 
take  place  are  qualitatively  in  accordance  with  the  facts,  it  is 
believed  that  the  admitted  defect  is  of  minor  importance. 

Before  proceeding  to  explain  the  method  in  applying  it  to  our 
2500-kva.  design,  let  us  bring  together  the  leading  data  which  we 
have  now  worked  out.  This  is  done  in  the  following  specification: 

SUMMARY  OF  THE  NORMAL  RATING  OF  THE  DESIGN 

Number  of  poles 8 

Output  at  full  load  in  kva 2500 

Corresponding  power-factor  of  external  load 0 . 90 

Corresponding  output  in  kw 2250 

Speed  in  r.p.m 375 

Periodicity  in  cycles  per  second 25 

Terminal  pressure  in  volts 12  000 

Number  of  phases 3 

Connection  of  phases Y 


Pressure  per  phase  ( =-  J .  .  .  .  .   6950 

THE  LEADING  DATA  OF  THE  DESIGN 

External  diameter  of  stator  core 2300  mm. 

Diameter  at  bottom  of  stator  slots 1894  mm. 

Internal  diameter  of  stator  (D) 1780  mm. 

External  diameter  of  rotor 1744  mm. 

Diameter  at  bottom  of  pole  shoes 1580  mm. 

External' diameter  of  magnet  yoke 1180  mm. 

Internal  diameter  of  magnet  yoke 928  mm. 

Polar  pitch  (T) 700  mm. 

Gross  core  length  (Xgr) 1180  mm. 

Number  of  vertical  ventilating  ducts 20 

Width  each  duct . 15  mm. 

Per  cent  insulation  between  laminations 10  per  cent 

Net  core  length  (1180-300)  X0.90  (Xn) 792  mm. 

Length  pole  shoe  parallel  to  shaft 1140  mm. 

Pole  arc 420  mm. 

Area  of  pole  face  (114X42) 4790  sq.cm. 

Extreme  length  magnet  core  parallel  to  shaft. .  .  860  mm. 

Extreme  width  magnet  core 260  mm. 

Area  of  cross-section  of  magnet  core 2080  sq.cm. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR          41 

Length  of  yoke  parallel  to  shaft  ...............  1200  mm. 

Radial  depth  of  yoke  ........................     126  mm. 

Cross-section  of  yoke  ................  .  .......  1510  sq.cm. 

Number  of  stator  slots  ......................     120 

Number  stator  slots  per  pole  per  phase 


Depth  slot  .................................       57  mm. 

Width  slot  ................................       24  mm. 

Width  slot  opening  .........................       12  mm. 

Sketches  of  the  design  have  already  been  given  in  Figs.  20 
and  21.  These  preliminary  sketches  show  wide-open  slots.  Let 
us,  however,  employ  slots  with  an  opening  of  only  12  mm.  in 
accordance  with  the  above  tabulated  specification. 

The  stator  winding  consists  of  10  conductors  per  slot.  The 
bare  dimensions  of  each  conductor  are  12  mm.  X  3.3  mm.,  and 
the  10  conductors  are  arranged  one  above  the  other  as  already 
indicated  in  Fig.  1  on  page  11.  The  mean  length  of  one  armature 
turn  is  656  cm.  The  winding  is  of  the  type  which  we  have  termed 
a  "  half-coiled  "  winding.  That  is  to  say,  only  half  the  field 
poles  have  opposite  to  them,  armature  coils  belonging  to  any 
one  phase.  A  diagram  of  one  phase  of  a  typical  half-coiled 
winding  for  a  6-pole  machine  has  been  given  in  Fig.  15  on  p.  20. 

L  /  Q\ 

and.  is  seen  to  havef—  )  =  three  coils  per  phase.     Since  our  2500 

kva.  machine  has  8  poles,  there  are  four  coils  in  each  phase. 
Each  side  of  each  coil  comprises  the  contents  of  five  adjacent  slots. 
Since  each  slot  contains  10  conductors,  there  are  (5  X  10  =  )  50  turns 
per  coil,  and  consequently  (4X50  =  )  200  turns  in  series  per  phase. 
Denoting  by  T  the  number  of  turns  in  series  per  phase  we  have 

^  =  200. 

In  Fig.  24  is  given  a  winding  diagram  for  one  of  the  three 
phases,  and  in  Fig.  25,  is  given  a  winding  diagram  containing  all 
three  phases. 

/1  9  000\ 
For  a  pressure  of  (  -       -  )  =  6950  volts  per  phase  on  open 

\   v  3  / 

circuit,  the  armature  flux  per  pole  (denoted  by  M),  is  obtained 
as  follows: 


42         POLYPHASE  GENERATORS  AND  MOTORS 


We  may  take  the  leakage  factor  as  1.15.     Consequently  the 
flux  in  the  magnet  core  and  yoke  is: 

1.15X31.3  =  36.0  megalines. 


FIG.  24. — Winding  Diagram  for  one  of  the  Three  Phases  of  the  2500-kva. 
three-phase  Alternator. 

The  estimation  of  the  no-load  saturation  curve  has  previovsly 
been  carried  out  in  earlier  sections.  A  summary  of  the  component 
and  resultant  magnetomotive  forces  (mmf.)  for  6000,  6950,  and 
7500  volts,  is  given  in  the  following  table: 


6000  Volts. 

6950  Volts. 

7500  Volts. 

Air-gap 

10  000  ats. 

11  560  ats. 

12  500  ats 

Teeth  

90  " 

330   " 

720   " 

Magnet  core  
Magnet  yoke  .... 
Stator  core  

770  " 
200  " 
140  " 

2000   " 
320   " 
190  " 

5300   " 
420   " 
230   " 

Total  mmf  

11  200  ats. 

14  400  ats. 

19  170  ats. 

CALCULATIONS  FOR  2500-KVA.  GENERATOR       43 

A  no-load  saturation  curve  passing  through  these  three  points 
has  been  given  in  Fig.  23,  on  page  39. 

The  Armature  Interfering  mmf.     The  current  per  phase  at 
rated  load  of  2500  kva.  is: 

2500000 
1  ~  3X6950  "12U' 


FIG.  25. — Complete  Winding  Diagram  for  all  Three  Phases  of  the  2500-kva. 

Generator. 


We  have  seen  that  there  are  200  turns  in  series  in  each  phase. 
Consequently  there  are: 


and 


200     or  ±  , 

-Q-  =  25  turns  per  pole  per  phase 

o 

25X120  =  3000  (rms.)  ats.  per  pole  per  phase. 


It  has  already  been  stated   that  many  designers  resort  to 
theoretical  reasoning  in  ascertaining  from  the  mmf.  of  one  phase, 


44         POLYPHASE  GENERATORS  AND  MOTORS 

the  resultant  mmf.  exerted  by  the  three  phases.  But,  in  practice, 
the  distribution  of  the  stator  and  rotor  windings,  the  ratio  of  the 
pole  arc  to  the  pitch,  and  other  details  of  the  design,  so  complicate 
the  matter  as  considerably  to  invalidate  any  theoretical  deduc- 
tions. But  working  backward  from  a  very  large  collection  of 
experimental  observations,  the  conclusion  is  reached  that  inde- 
pendently of  the  various  relative  dispositions  of  the  winding  and 
of  other  features  of  the  design,  results  consistent  with  practice 
are  obtained  by  taking  the  resultant  mmf.  of  the  three  phases  as 
equal  to: 

2.4  times  the  mmf.  of  each  phase. 

Consequently  for  our  design,  we  have,  at  full  load,  an  armature 
mmf.  of: 

2.4X3000  =  7200  ats. 

It  is  only  at  zero  power-factor  that  these  armature  ats.  have 
the  same  axis  as  the  field  ampere-turns.  If,  when  the  power- 
factor  of  the  external  load  is  zero,  the  output  is  120  amperes, 
then,  if  the  current  is  lagging,  the  resultant  mmf.  acting  to  send 
flux  round  the  magnetic  circuit  is  obtained  by  subtracting  7200 
ats.  (the  armature  mmf.)  from  the  excitation  on  each  field  pole. 
If,  with  the  power-factor  again  equal  to  zero,  the  current  is  lead- 
ing, then  the  resultant  mmf.  acting  to  send  flux  round  the  magnetic 
circuit  is  obtained  by  adding  7200  ats.  (the  armature  mmf.)  to  the 
excitation  on  each  field  pole.  For  this  same  current  of  120 
amperes,  but  at  other  than  zero  power-factor,  the  armature  mmf. 
does  not  affect  the  resultant  mmf.  to  so  great  an  extent. 

Later  we  shall  consider  a  method  of  determining  the  extent 
of  the  influence  of  the  armature  mmf.  when  the  power-factor 
is  other  than  zero.  It  follows  as  a  consequence  of  the  preceding 
explanations  that  the  armature  mmf.  will  exert  a  less  disturbing 
influence  on  the  terminal  pressure  the  greater  the  mmf.  provided 
on  the  field  spools,  and  that  for  a  given  required  closeness  of 
pressure  regulation  the  higher  the  armature  mmf.,  the  higher  must 
also  be  the  field  mmf. 

The  modern  conception  of  preferable  conditions  is  not  based 
on  such  close  inherent  pressure  regulation  as  was  formerly  con- 
sidered desirable.  The  alteration  in  conceptions  in  this  respect 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       45 

does  not,  however,  decrease  the  importance  of  having  at  our  dis- 
posal means  for  accurately  estimating  the  excitation  required 
under  various  conditions  of  load  as  regards  pressure,  power-factor 
and  amount. 

The  Field  Excitation  Required  with  Various  Loads.  The 
required  excitation  is  chiefly  dependent  upon  three  factors : 

1.  The  no-load  saturation  curve  of  the  machine; 

2.  The  armature  strength  in  ats.  per  pole; 

3.  The  inductance  of  the  armature  winding. 

In  our  2500-kva  three-phase  machine,  the  armature  strength 
at  rated  load  is  equal  to  7200  ats. 

The  Position  of  the  Axis  of  the  Armature  mmf .  If  the  arma- 
ture winding  had  no  inductance,  then  for  an  external  load  of 
unity  power-factor,  the  axis  of  the  armature  magnetomotive 
force  would  be  situated  just  midway  between  two  adjacent 
poles;  that  is  to  say,  there  would  be  no  direct  demagnetization. 
At  the  other  extreme,  namely  for  the  same  current  output  at  zero 
power-factor,  the  axis  of  armature  demagnetization  would  cor- 
respond with  the  field  axis.  The  two  cases  are  illustrated  dia- 
grammatically  in  Figs.  26  and  27.  In  our  machine,  when  loaded 
with  full-load  current  of  120  amperes  at  zero  power-factor,  the 
demagnetization  would  amount  to  7200  ats.  and  this  demagnet- 
ization could  only  be  offset  by  providing  7200  ats.  on  each  field 
pole.  For  power-factors  between  1  and  0,  the  axis  of  armature 
demagnetization  would  be  intermediate,  as  indicated  diagram- 
matically  in  Fig.  28. 

But  we  are  not  concerned  with  imaginary  alternators  with 
zero-inductance  armature  windings,  but  with  actual  alternators. 
In  actual  alternators,  the  armature  windings  have  considerable 
inductance.  At  this  stage  we  wish  to  determine  the  inductance 
of  the  armature  windings  of  our  2500-kva  alternator. 

The  Inductance  of  a  i-turn  Coil.  Let  us  first  consider  a  single 
turn  of  the  armature  winding  before  it  is  put  into  place  in  the  stator 
slots.  If  we  were  to  send  one  ampere  of  continuous  electricity 
through  this  turn,  how  many  magnetic  lines  would  be  occasioned? 
If  the  conductor  were  large  enough  to  practically  fill  the  entire 
slot,  then  with  the  dimensions  employed  in  modern  alternators, 
the  general  order  of  magnitude  of  the  flux  occasioned  may  be 
ascertained  on  the  basis  that  some  0.3  to  0.9  of  a  line  would  be 


46         POLYPHASE  GENERATORS  AND  MOTORS 


Direction  of  dotation 


FIG.  26 — Diagrammatic  Representation  of  Relative  Positions  of  Axes  of  Field 
mmf .  and  Armature  mmf .  for  a  Load  of  Unity  Power-factor  Neglecting 
Armature  Inductance. 


Direction  of  Rotation 


FIG.  27. — Diagrammatic  Representation  of  Relative  Positions  of  Axes  of  Field 
mmf.  and  Armature  mmf.  for  a  Load  of  Zero  Power-factor. 


Direction  of  Rotation 


FIG.  28. — Diagrammatic  Representation  of  Relative  Positions  of  Axes  of  Field 
mmf.  and  Armature  mmf.  for  a  Load  of  Intermediate  Power-factor. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR 

linked  with  every  centimeter  of  length  of  the  turn.  Taking  the 
mean  value  of  0.6  line  per  cm.,  then  since  in  our  design  the 
length  of  a  turn  is  656  cm.,  the  flux  occasioned  by  1  amp.  of 
continuous  electricity  is 

656X0.6  =  394  lines. 

The  inductance  (expressed  in  henrys)  of  a  1-turn  coil  is  equal 
to  10~8  times  the  number  of  lines  linked  with  the  turn  when  1  amp. 
of  continuous  electricity  is  flowing  through  the  turn.  Conse- 
quently the  inductance  is,  in  this  case,  equal  to 

10~8X  394  =  0.00000394  henry. 

The  Inductance  of  a  Coil  with  More  than  One  Turn.  The 

inductance  of  any  coil  is  equal  to  the  product  (when  1  amp.  of 
continuous  electricity  is  flowing  through  the  coil),  of  the  flux 
linked  with  the  coil  and  the  number  of  turns  in  the  coil.  This 
definition  is  framed  on  the  assumption  that  the  entire  flux  is  linked 
with  the  entire  number  of  turns.  Where  this  is  not  the  case, 
appropriate  factors  must  be  employed  in  order  to  arrive  at  the 
correct  result. 

In  a  two-turn  coil,  the  mmf .  is,  when  a  current  of  1  amp.  of 
continuous  electricity  is  flowing  through  the  coil,  twice  as  great 
as  in  a  one-turn  coil  of  the  same  dimensions.  Consequently  for 
a  magnetic  circuit  of  air,  the  flux  will  also  be  twice  as  great, 
since  in  air  the  flux  is  directly  proportional  to  the  mmf.  occa- 
sioning it.  But  since  this  doubled  flux  is  linked  with  double  the 
number  of  turns,  the  total  linkage  of  flux  and  turns  is  four  times 
as  great.  In  other  words,  the  inductance  increases  as  the  square 
of  the  number  of  turns. 

In  Fig.  24,  it  has  been  shown  that  the  winding  of  any  one 
phase  of  our  eight-pole  machine  is  composed  of  four  coils  in  series. 
Let  us  first  consider  one  of  those  four  coils.  Each  side  com- 
prises the  contents  of  five  slots.  Since  there  are  ten  conductors 
per  slot,  we  see  that  we  are  dealing  with  a  fifty-turn  coil.  On  the 
sufficient  assumption  that  the  incomplete  linkage  of  flux  and  turns 
is  provided  for  by  calculating  from  the  basis  of  only  0.5  line  per 
centimeter  of  length,  instead  of  from  the  value  of  0.6  line  per  centi- 


48         POLYPHASE  GENERATORS  AND  MOTORS 

meter  of  length  which  we  employed  when  dealing  with  the  one- 
turn  coil,  we  obtain  for  the  inductance  the  value: 

502X^1x0.00000394  =  0.0082  henry. 

The  Inductance  and  Reactance  of  One  Phase.  The  winding 
of  one  phase  comprises  four  such  coils  in  series,  and  consequently 
we  have: 

Inductance  per  phase  =  4  X  0.0082  =  0.0328  henry 
The  reactance  is  obtained  from  the  formula: 
Reactance  (in  ohms) 


where  the  periodicity  in  cycles  per  second  is  denoted  by  ^  and  the 
inductance  in  henry  s  by  1.  We  consequently  have: 

Reactance  per  phase  =  6.28X25X0.0328  =  5.  15  ohms. 

The  Reactance  Voltage  per  Phase.  For  our  machine  the 
full-load  current  per  phase  is  120  amperes.  Consequently  when 
carrying  full-load  current  we  have: 

Reactance  voltage  per  phase  =  120X5.  15  =  618  volts. 

The  Inductance  and  Reactance  of  Slot-embedded  Windings. 

But  up  to  this  point  we  have  considered  that  throughout  their 
length  the  windings  are  surrounded  by  air.  In  reality  the  wind- 
ings are  embedded  in  slots  for  a  certain  portion  of  their  length. 
For  this  embedded  portion  of  their  length,  the  flux,  in  lines  per 
centimeter  of  length,  set  up  in  a  one-turn  coil  when  one  ampere 
of  continuous  electricity  flows  through  it,  is  considerably  greater 
than  for  those  portions  of  the  coil  which  are  surrounded  by  air. 
Suitable  values  may  be  obtained  from  Table-  9  : 

TABLE  9.    DATA  FOR  ESTIMATING  THE  INDUCTANCE  OF  THE  EMBEDDED  LENGTH. 

No.  of  Lines 

per  cm. 
Concentrated  windings  in  wide-open,  straight-sided  slots  ..........       3  to  6 

Thoroughly  distributed  windings  in  wide-open,  straight-sided  slots    1.5  to  3 
Concentrated  windings  in  completely-closed  slots  ..........  .......    7  to  14 

Thoroughly-distributed  windings  in  completely-closed  slots  ........       3  to  6 

Partly  distributed  windings  in  semi-closed  slots  ...................    about  5 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       49 


In  order  to  illustrate  the  sense 
centrated,"  "  thoroughly-distrib- 
uted" and  " partly-distributed" 
windings  are  employed  in  the 
above  table,  the  three  winding 
diagrams  in  Figs.  29,  30  and  31 
have  been  prepared.  Evidently 
for  the  windings  of  any  one 
phase  of  our  machine,  the 
value  of  5  lines  per  cm.  of 
embedded  length  is  sufficiently 
representative. 

The  embedded  portion  of  the 
length  of  a  turn  is  equal  to  twice 
the  net  core  length.  For  our 
machine  we  have : 


in  which    the  terms  "  con- 


FIG.  29.— Concentrated  Winding. 


Embedded  length  =  2X79  =  158  cm.; 

Mean  length  of  a  turn  =  656  cm. 

"  Free  "  length  (i.e.,  the  portion  in  air)  =  656  - 158  =  498  cm. 


•ury~ui^n^njij"yru^^ 


FIG.  30, 
Thoroughly  Distributed  Winding. 


FIG.  31. 
Partly  Distributed  Winding. 


We  have  calculated  the  inductance  which  our  coil  would  have 
if  the  entire  656  cm.  of  its  length  were  surrounded  by  air  (i.e., 
were  "  free  "  length).  We  can  now  readily  obtain  the  value  of 


50         POLYPHASE  GENERATORS  AND  MOTORS 

that  part  of  the  inductance  which  is  associated  with  the   actual 
"  free  "  length.     It  amounts  to: 


iX  0.0328  =  0.0248  henry. 
656 

The  inductance  of  the  "  embedded  "  length  is 

j^XJ^X  0.0328  =  0.0790  henry. 


The  total  inductance  per  phase  is 

0.0248+0.0790  =  0.104  henry. 
It  is  interesting  to  note  that 

0.0248. 


0.104 


X 100  =  23.8  per  cent 


of  the  total  inductance,  is,  in  the  case  of  this  particular  machine, 
associated  with  the  end  connections. 

Our  estimate  of  the  inductance  has  been  so  seriously  interrupted 
by  explanatory  text  that  it  is  desirable  to  set  it  forth  again  in  a 
more  orderly  form,  and  taking  each  step  in  logical  order  : 

Mean  length  of  turn  ..................   656  cm. 

"  Free  "  length  ......................  498  cm. 

"  Embedded  "  length  ..................   158  cm. 

Flux  per  ampere-turn  per  j  0.5  line  for  "  free  "  length, 

centimeter  of  length      [  5.0  lines  for  "embedded  "  length,  J 

(249  lines  for  "  free  "  length, 
Flux  per  ampere-turn  j  ?go  ^  for   M  embedded  „   length< 


Total  flux  per  ampere-turn  (  =  249+790=)  1040  lines. 

Number  of  turns  in  one  phase  per  pair  of  poles  (i.e., 
per  coil)  .................................  50 


CALCULATIONS  FOR  2500-KVA.  GENERATOR   51 

Inductance  of  one  coil  (1040X502X10~8  =  )  ......   0.0260  henry 

Number  of  coils  (also  pairs  of  poles)  per  phase  .....  4 

Inductance  of  one  phase  (4X0.0260  =  )    ...........  0.104  henry 

Reactance  of  one  phase  at  25  cycles  (6.28X25X0.104  =  )  16.3  ohms 
Reactance  voltage  of  one  phase  at  25  cycles  and  120 

amperes  (120X16.3)  =     .....................     1960  volts 

Physical  Conditions  Corresponding  to  this  Value  of  the 
Reactance  Voltage.  This  value  of  1960  volts  for  the  reactance 
voltage,  is  of  the  order  of  the  value  which  we  should  obtain 
experimentally  under  the  following  conditions: 

Twenty-five-cycle  current  is  sent  into  the  stator  windings  from 
some  external  source,  while  the  rotor  (unexcited)  ,  is,  by  means  of  a 
motor,  driven  at  the  slowest  speed  consistent  with  steady  indica- 
tions of  the  current  flowing  into  the  three  branches  of  the  stator 
windings.  Under  these  conditions,  some  1960  volts  per  phase 
would  be  found  to  be  necessary  in  order  to  send  120  amperes  into 
each  of  the  three  windings. 

Theta  (0)  and  Its  Significance.  The  value  of  the  reactance 
voltage  thus  determined,  enables  us  to  ascertain  the  angular 
distance  from  mid-pole-face  position  at  which  the  current  in  the 
stator  windings  passes  through  its  crest  value. 

Let  this  angle  be  denoted  by  0.  For  a  load  of  unity  power- 
factor,  0  is  the  angle  whose  tangent  is  equal  to  the  reactance 
voltage  divided  by  the  phase  voltage.  Thus  we  have: 

!  reactance  voltage 
6  =  tan    J  —  r—    —  . 

phase  voltage 

The  conception  of  0  may  possibly  be  made  clearer  by  stat- 
ing that  it  represents  the  angular  distance  by  which  the  center 
of  a  group  of  conductors  belonging  to  one  phase  has  traversed 
beyond  mid-pole-face  position  when  the  current  in  these  conduc- 
tors reaches  its  crest  value. 

Theta  at  Unity  Power-factor.  For  our  example  we  have 
(for  unity  power-factor)  : 

tan~1  0.282  =  15.9°.* 


*  In  making  calculations  of  the  kind  explained  in  this  Chapter,  the  Table 
of  sines,  cosines  and  tangents  in  Appendix  III.  will  be  found  useful. 


52 


POLYPHASE  GENERATORS  AND  MOTORS 


The  diagram  is  shown  in  Fig.  32.     Strictly  speaking,  we  ought 
to  take  into  account  in  the  diagram,  the  IR  drop  in  the  armature. 


FIG.  32.— ^Diagram  Relating  to  the  Explanation  of  the  Nature  and  Significance 
of  the  Angle  Theta  (6). 

The  resistance  per  phase  (at  60°  Cent.),  is  0.685  ohm.     Conse- 
quently for  the  full-load    current  of  120  amperes,  we  have: 

IR  drop  =  120X0.685  =  82    volts. 

The    corrected    diagram    (taking  into  account  the  IR  drop), 
is  shown  in  Fig.  33.     In  this  diagram  we  have: 

6-tBa-»g^=tan->^»taa-»0^8*i5^ 


FIG.  33. — More  Exact  Diagram  for  Obtaining  0. 

Relation   between    Theta   and   the    Armature    Interference. 

The  armature  demagnetization  for  any  value  of  6  is  obtained  by 
multiplying  the  armature  strength  by  sin  0. 
For  120  amperes  the  armature  strength  is : 


7200  ats. 


We  also  have: 


sin  6  =sin  15.8°  =  0.270. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       53 

Under  these  conditions  (120  amperes  output  at  unity  power- 
factor  of  the  external  load),  the  armature  mmf.  is  equal  to: 


0.270  X  7200  =  1940  ats. 


The  Hypothenuse  of  the  6-Triangle  Has  no  Physical  Exist- 
ence. It  is  desirable  to  lay  strong  emphasis  on  the  fact  that 
the  vector  sum  of  7032  volts  and  1960  volts  does  not  represent 
an  actually-existing  internal  pressure  corresponding  to  an  actual 
flux  of  magnetic  lines.  The  quantity  which,  earlier  in  this 
chapter,  has  been  termed  the  "  reactance  voltage  ",  is  made  up 
of  two  parts,  associated  respectively  with  the  "  embedded " 
length  and  with  the  "  free  "  length.  While  the  portion  associated 
with  the  "  embedded  "  length  manifests  itself  in  distortion  of 
the  magnetic  flux,  the  portion  associated  with  the  "  free  "  length 
acts  in  the  same  manner  as  would  an  equal  inductance  located 
in  an  independent  inductance  coil  connected  in  series  with  a  non- 
inductive  alternator.  (More  strictly,  it  is  only  that  portion  of 
the  "  free  "  length  which  is  associated  with  the  end  connections 
which  should  be  thus  considered,  and  the  portion  associated 
with  the  ventilating  ducts  should  be  placed  in  a  different  cate- 
gory. But  in  practice  the  small  margin  provided  by  taking  the 
entire  "  free  "  length,  is  desirable.) 

We  have  seen  that  the  inductance  of  the  "  free  "  length  of 
the  windings  of  our  2500-kva.  machine  is  23.8  per  cent  of  the  total 
inductance ; 

0.238X1960  =  466  volts. 

The  True  Internal  Pressure  and  Its  Components.  The  three 
components  of  the  total  internal  pressure  of  our  machine,  when 
the  external  load  is  120  amperes  at  unity  power-factor,  are,  per 
phase : 

Phase  pressure .  .  . 6950  volts 

IRdrop 82    " 

Reactance  drop 466    ' ' 


54         POLYPHASE  GENERATORS  AND  MOTORS 

When  these  are  correctly  combined,  as  shown  in  Fig.  34,  the 
internal  pressure  is  ascertained  to  be: 


V?0322+4662  =  7050  volts. 

The  influence  of  the  reactance  voltage  is  thus  (for  these  par- 
ticular conditions  of  load),  practically  negligible,  so  far  as  concerns 
occasioning  an  internal  pressure  appreciably  exceeding  the  result- 
ant of  the  terminal  pressure  and  the  IR  drop. 


FIG.   34. — Pressure   Diagram  Corresponding  to   6950  Terminal  Volts  and 
120  Amperes  at  Unity  Power-factor. 

Total  mmf .  Required  at  Full  Load  and  Unity  Power-factor. 
From  the  no-load  saturation  curve  in  Fig.  23  on  p.  39,  we  see 
that  we  require: 

15200  ats. 

to  overcome  the  reluctance  of  the  magnetic  circuit  when  the 
internal  pressure  is: 

7050  volts. 

• 

We  require  further: 

1940  ats. 

to  offset  the  armature  demagnetization  for  these  conditions  of 
load  (120  amperes  at  unity  power-factor).  Consequently  we 
require  a  total  mmf.  per  field  spool,  of: 

15  200+1940  =  17140  ats. 

That  is  to  say,  for  full-load  conditions  (6950  volts  per  phase 
and  120  amperes  at  unity  power-factor),  we  require  an  excitation 
of: 

17  140  ats. 

The  Inherent  Regulation  at  Unity  Power-factor.  We  can 
now  ascertain  from  the  saturation  curve  the  value  to  which  the 


CALCULATIONS  FOR  2500-KVA.  GENERATOR        55 

pressure  will  rise,  when,  maintaining  constant  this  excitation  of 
17  140  ats.,  we  decrease  the  load  to  zero.  We  find  the  value  of 
the  pressure  to  be: 

7350  volts. 

Thus  the  pressure  rise  occurring  when  the  load  is  decreased 
to  zero,  is: 

/^-6950xxtAft    \KQ 

(      6950       X100  =  J5.8  per  cent. 

This  is  expressed  by  stating  that  at  unity  power-factor  the  inherent 
regulation  is: 

5.8  per  cent. 


ESTIMATION  OF  SATURATION  CURVE  FOR  UNITY  POWER 
FACTOR  AND   120  AMPERES 

Let  us  now  proceed  to  calculate  values  from  which  we  can 
plot  a  load  saturation  curve  extending  from  a  pressure  of  0  volts 
up  to  a  pressure  of  7500  volts  for  an  external  load  of  120  amperes 
at  unity  power-factor.  We  already  have  one  point;  namely: 

17  140  ats.  for  6950  volts. 

For  this  unity  power-factor,  120-ampere  saturation  curve,  the 
terminal  pressure  will  be  varied  from  0  up  to  a  phase  pressure 
of  —  say  —  7500  volts  while  the  current  is  held  constant  at  120 
amperes.  The  diagrams  for  obtaining  0  fop  7500,  5000,  2500  and 
0  volts  are  shown  in  Fig.  35.  For  these  four  cases  we  have: 


-1  0.258  =  14.5°  sin  14.5°  =  0.250 

2.  0  =  tan~1          =  tan-1  0.386  =  21.1°  sin  21.1°  =  0.360 

oUo-^ 

3.  6  =  tan-1  ~?  =  tan-1  0.758  =  37.1°  sin  37.1°  =  0.605 


4.      6  =  tan-1  —  ^  =tan~1  23.9  =  87.5°          sin  87.5°  =  0.999 


56         POLYPHASE  GENERATORS  AND  MOTORS 


7500 


5000 


2500 


,82 


FIG.  35. — Theta  Diagrams  for  120  Amperes  at  Unity  Power-factor. 

Since  the  armature  current  is,  in  all  four  cases,  120  amperes, 
the  armature  strength  remains  7200  ats.  The  armature  demag- 
netization amounts,  in  the  four  cases,  to : 

1.  0.250X7200  =  1800  ats.; 

2.  0.360X7200  =  2590  " 

3.  0.605X7200  =  4350  " 

4.  0.999X7200  =  7200  " 


CALCULATIONS  FOR  2500-KVA.  GENERATOR        57 


2500 


FIG.  36. — Pressure  Diagrams  for  120  Amperes  at  Unity  Power-factor. 


The  Armature  Reaction  with  Short-circuited  Armature. 
It  is  interesting  to  note  that  in  the  last  diagram  in  Fig.  35, 
i.e.,  in  the  diagram  relating  to  zero  terminal  pressure  (short- 
circuited  armature)  the  angle  0  is  practically  90°.  Con- 
sequently the  armature  reaction  with  short-circuited  armature, 
is  practically  identical  with  the  armature  strength  expressed 
in  ampere-turns  per  pole. 

The  Required  Field  Excitation  for  Each  Terminal  Pressure. 
The  field  excitation  at  each  pressure,  comprises  two  components. 
The  first  of  these  components  must  be  equal  to  the  armature 
demagnetization  (in  order  to  neutralize  it),  and  the  second  com- 
ponent must  be  of  the  right  amount  to  drive  the  required  flux 
through  the  magnetic  circuit  in  opposition  to  its  magnetic  reluct- 
ance. This  latter  value  may  be  obtained  from  the  no-load 
saturation  curve  in  Fig.  23  (on  p.  39),  and  must  correspond  to  the 


58 


POLYPHASE  GENERATORS  AND  MOTORS 


four  pressures    obtained   from   the   four   diagrams    in   Fig.  36- 
These  four  pressures  are: 


1.     V75822+4662  =  7590  volts. 


2.     V50822+4662  = 


3.     \/25822+4662  = 


4.     V822+4662  = 


The  saturation  ats.  for  these  four  pressures  are  found  from 
Fig.  23  (on  p.  39),  to  be  as  follows: 


1. 

Pressure 

7590 

Sat.  Ats 

22000 

2. 

5100 

9100 

3. 

2630 

4700 

4. 

476 

820 

The  derivation  of  the  total  required  mmf .  is  arranged  below 
in  tabular  form : 


Terminal 
Pressure. 

Saturation 
Ats. 

Ats.for  Offsetting 
Armature 
Demagnetization. 

Total  Required 

Ats. 

1 

7500 

22000 

1800 

23800 

2 

5000 

9100 

2950 

11690 

3 

2500 

4700 

4350 

9050 

[4 

0 

820 

7200 

8020 

The  unity-power-factor,  120-ampere,  saturation  curve,  thus 
derived,  is  plotted  in  Fig.  37,  where  the  no-load  saturation  curve 
is  also  reproduced  from  Fig.  23. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       59 


7,000 


6,000 


|  5,000 

ri 


4,000 


2,000 


.1,000 


-cfy 


N^<ooQ033<oSo    gf  $  $  $  s 

mmf  per  Field  Spool  in  ats 
FIG.  37. — Saturation  Curves  for  7  =  120  and  (r  =  1.00  and  for  7=0. 


ESTIMATION  OF  SATURATION  CURVE  FOR  UNITY  POWER- 
FACTOR  AND   240  AMPERES 

Let  us  now  construct  a  saturation  curve  for  unity  power- 
factor  and  240  amperes  output,  i.e.,  for  twice  full-load  current. 

The  diagrams  of  Figs.  35  and  36  are  now  replaced  by  those 
of  Figs.  38  and  39. 

The  reactance  voltage  for  the  0-diagrams  of  Fig.  38  is  now 
twice  as  great  as  before,  since  the  current  is  now  240  amperes 
in  place  of  120  amperes. 

The  reactance  voltage  is  now: 

2X1960  =  3920  volts. 


The  7.R  drop  is  now : 


2X82  =  164  volts. 


60         POLYPHASE  GENERATORS  AND  MOTORS 


/164 

2 

3 

^\^^ 

1 

o 

n/^\^ 

CO 

$               \. 

/164 

^ 

r      ^^^ 

4  —  -__  ^_^7T2Q 

*                      _NW 

S|  — 

7500 

7500 

^ 

164 

•  3 

0 

> 

1 

164. 

I 

r—  ^^^ 

5000 

5000 

/164 

^ 

2 

3 

>• 

1 

164 

2500 

2500 

J164 

1 

• 

p> 

1 

/164. 

0 

. 

/ 

OT  ^j05 

FIG.  38.  —  Theta  Diagrams  for  /  =  240     FIG.  39.  —  Pressure  Diagrams  for 

andG  =  1.00.                                    /  =  240  and  G  =  1  .00. 

Consequently  : 

1             ft        tin  —1    """"        +nvl  —  1   H   X*]  1         <~>7   1  °               cnr»  ^71°        H  A^ft 

i.      u  —  tan      lyfttsA  —  tan      u.oii  —  44.1           sin^/.i   —  u.^oo 

2.      6  =  tan-1  Irl^tan-1  0.760  =  37.2°         sin  37.2°  =  0.605 
olb4 

oqorj 

3.      6  =  tan-1~^  =  tan-1  1.47   =55.7°        sin  55.7°  =  0.826 

oqor) 

4.     6  =  tan~1^^  =  tan-1  23.8   =87.5°        sin  87.5°  =  0.999 

CALCULATIONS  FOR  2500-KVA.  GENERATOR       61 
The  armature  strength  is  now; 

2x7200  =  14400ats. 

Consequently  in  the  four  cases,  we  now  have  for  the  armature 
demagnetizing  ats.  : 

1.  0.456X14400  =  6580  ats. 

2.  0.605X14400  =  8700    " 

3.  0.826X14400  =  11900  " 

4.  0.999X14400  =  14400  " 

The  internal  inductance  pressure  is  now: 
2X466  =  932  volts. 

The  four  internal  pressures  and  the  corresponding  saturation 
ats.  are: 


Internal  Pressures. 

1  .     V76642+9322  =  7720 

2.  V51642+9322  =  5250 

3.  V26642+9322 


4.     V  1642+9322  =  945 


Sat.  Ats. 

29  500 
9  350 
5150 
1700 


The  total  required  ats.  are  shown  in  the  last  column  of  the 
following  tabulated  calculation: 


Term.  Pres. 

Saturation 
Ats. 

Ats.  for  Offsetting 
Armature 
Demagnetization. 

Total  Required 
Ats. 

1 

7500 

29500 

6580 

36080 

2 

5000 

9350 

8700 

18050 

3 

2500 

5150 

11900 

17050 

4 

0 

1700 

14400 

16  100 

62 


POLYPHASE  GENERATORS  AND  MOTORS 


The  values   in    the  last  column  are  the  basis  for  the  unity 
power-factor,   240-ampere    saturation    curve  shown  in  Fig.  40. 


8,000 


c3       £J       N       c§      c§       eo      co       «>       «o      P 
mmf  per  Field  Spool,  in  ats  , 

FIG.  40. — Saturation  Curves  for  Various  Values  of  7  and  for  (r  =  1.00. 

The   unit}'   power-factor,  120-ampere   curve   is   also   reproduced 
from  Fig.  37  and  the  7=0  curve  from  Fig.  23. 

SATURATION  CURVES  FOR  POWER-FACTORS  OF  LESS  THAN 

UNITY 

Let  us  now  return  to  a  load  of  120  amperes,  but  let  the  power- 
factor  of  the  load  on  the  generator  be  0.90.  Let  us  estimate  the 
required  mmf.  under  these  conditions,  for  terminal  pressures  of 
7500,  5000,  2500,  and  0  volts,  and  then,  from  these  four  results, 
let  us  plot  a  0.90-power-f actor,  120-ampere,  saturation  curve. 

The  angle  0,  i.e.,  the  angle  by  which  the  conductors  have  passed 
mid-pole-face  position  when  carrying  the  crest  current,  is  now 
obtained  as  follows: 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       63 


When  the  power-factor  of  the  external  load  is  0.90,  the  current 
lags  behind  the  terminal  pressure  by  26.0°,  since  cos  26°  =  0.90. 

The  0-diagram  for  120  amperes  and  a  terminal  pressure  of 
6950  volts,  is  now  as  shown  in  Fig.  41.  The  entire  object  of  this 
diagram  is  to  obtain  the  angle  0,  i.e.,  the  angle  by  which  the  con- 


FIG.  41. — Theta  Diagram  for  7  =  120      FIG.  42. — Pressure  Diagram  for 
and  G  =  0.90.  7  =  120  and  G=0.90. 

ductors  have  passed  mid-pole-face  position  when  the  current  is 
at  its  crest  value. 

AB  =  BCsm2Q° 

=  6950X0.438  =  3040 
AC=  0.90X6950  =  6250 

=AB+BE 
~ AC+DE 

^3040+1960 
"6250+  82 

5000 


Therefore 


6332 
=  0.790. 

6  =  38.3° 
sin  38.3°  =  0.620. 


Therefore : 

Armature  demagnetizing  ats.  =  0.620X7200  =  4450  ats. 


64         POLYPHASE  GENERATORS  AND  MOTORS 


The  diagram  for  obtaining  the  internal  pressure  is  shown  in 
Fig.  42.  By  scaling  off  from  this  diagram,  we  find  that  the  internal 
pressure  is  7250  volts.  From  the  no-load  saturation  curve  we  find : 

Saturation  mmf.  for  7250  volts  =  16  700  ats. 

Thus  the  total  required  mmf.  for  a  phase  pressure  of  6950 
volts  with  a  load  of  120  amperes  at  a  power-factor  of  0.90,  is: 

4450+16  700  =  21  150  ats. 

For  loads  of  other  than  unity  power-factor,  the  most  expedi- 
tious method  of  arriving  at  the  results  is  usually  that  by  graphical 
constructions.  In  the  chart  of  Fig.  43  which  relates  to  the 
graphical  derivation  of  the  saturation  curve  for  120  amperes  at 
0.90  power-factor,  the  diagrams  in  the  right-hand  column  relate  to 
the  determination  of  the  internal  pressure.  The  first,  second, 
third  and  fourth  horizontal  rows  relate  respectively  to  the  diagrams 
for  phase  pressures  of  7200,  5000,  2500  and  0  volts. 

The  left-hand  vertical  row  of  diagrams  relates  to  the  construc- 
tions for  the  determination  of  6  for  these  four  terminal  pressures. 

From  the  internal-pressure  diagrams  in  Figs.  42  and  43  and 
from  the  no-load  saturation  curve  in  Fig.  23  we  find: 


Phase 
Pressure. 

Internal 
Pressure. 

Saturation 
Ats. 

0 

472 

1000 

2500 

2800 

5000 

5000 

5290 

9300 

6950 

7250 

16700 

7200 

7500 

19200 

From  the  0  diagrams  wre  obtain  the  following  results : 


Phase 
Pressure. 

Tan 

e. 

Angle 
6. 

Sin 
0. 

Ampere-turns 
Required  to 
Offset  Armature 

Demagnetization. 

0 

24 

87.5° 

0.999 

7200 

2500 

1.35 

53.5° 

0.804 

5780 

5000 

0.918 

42.5° 

0.675 

4850 

6950 

0.790 

38.3° 

0.620 

4450 

7200 

0.785 

38.1° 

0.617 

4440 

CALCULATIONS  FOR  2500-KVA.  GENERATOR       65 


Theta  Diagrams 


Pressure  Diagrams 


FIG.  43.— Theta  and  Pressure  Diagrams  for  7  =  120  and  0  =  0.90. 

We  are  now  in  a  position  to  obtain  the  total  ats.  for  each  value 
of  the  phase  pressure.     The  steps  are  shown  in  the  following  table : 


Phase  Pressure. 

Saturation  Ats. 

Ats.  Required  to 
Offset  Armature 
Demagnetization. 

Total  Ats.  for 
120  Amp.  and  0.90 
Power-factor. 

0 

1  000 

7200 

8200 

2500 

5000 

5780 

10780 

5000 

9300 

4850 

14150 

6950 

16  700 

4450 

21  150 

7200 

19200 

4440 

23640 

66        POLYPHASE  GENERATORS  AND  MOTORS 

These  values  for  120  amperes  at  0.90  power-factor  and  those 
previously  obtained  for  120  amperes  at  unity  power-factor,  give 
us  the  two  load-saturation  curves  plotted  in  Fig.  44.  We  see 
that  for  a  phase  pressure  of  6950  volts,  when  the  current  is  120 
amperes  and  at  0.90  power-factor,  the  required  excitation  is 
21  150  ats.  From  the  no-load  saturation  curve  we  find  that  an 


8,000 


7,000 


I  I 

of    «?    «>    oo-    $    3    3    g    «?    §    g    g-    g    £   o- 

mmf  per  Field  Spool,  in  ats 

FIG.  44.— Saturation  Curves  for  7  =  120  and  for  (7  =  1.00  and  0.90. 

excitation  of  21  150  ats.  occasions,  at  no  load,  a  phase  pressure 
of  7600  volts 


7600-6950 
6950 


X 100  =  9.4. 


The  inherent  regulation  at  0.90  power-factor  is,  for  this 
machine,  9.4  per  cent.  In  other  words,  if,  for  an  output  of  120 
amperes  at  a  power-factor  of  0.90  we  adjust  the  excitation  to  such 
a  value  as  to  give  a  phase  pressure  of  6950  volts,  and  if,  keeping 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       67 

the  excitation  constant  at  this  value,  the  load  is  decreased  to 
zero,  the  pressure  will  rise  9.4  per  cent. 


ESTIMATION  OF  SATURATION   CURVE    FOR  120    AMPERES 
AT  A  POWER  FACTOR  OF  0.80 

Now  let  us  carry  through  precisely  similar  calculations  for 
120  amperes  at  a  still  lower  power-factor,  namely,  a  power-factor 
of  0.80.  We  shall  first  estimate  the  required  mmf.  (at  120  amperes 
and  0.80  power-factor)  for  phase  pressures  of  7200,  5000,  2500  and 
0  volts,  and  from  these  four  results  we  can  plot  the  required  satu- 
ration curve. 

We  have  the  relation; 


cos"1  0.80  =  37.0°. 


The  reactance  voltage  and  the  internal  IR  drop  remain  the 
same  as  in  the  diagrams  of  Fig.  43.  From  these  data  we  readily 
arrive  at  the  diagrams  of  Fig.  45. 

From  the  internal-pressure  diagrams  in  Fig.  45  and  from  the 
no-load  saturation  curve  in  Fig.  23,  we  arrive  at  the  following 
results : 


Phase 
Pressure. 

Internal 
Pressure. 

Saturation 
Ats. 

0 

472 

1000 

2500 

2850 

5100 

5000 

5330 

9400 

7200 

7560 

20000 

From  the  6  diagrams  in  Fig.  45  we  obtain  the  following  results ; 


Phase  Pressure. 

0. 

Sin  6. 

Ats  Required 
to  Offset  Armature 
Demagnetization. 

0 

87.5° 

0.999 

7200 

2500 

60.2° 

0.868 

6250 

5000 

51.0° 

0.777 

5600 

7290 

47.5° 

0.737 

5300 

68 


POLYPHASE  GENERATORS  AND  MOTORS 


-82 


,82 


?=60.2 


Theta  Diagram 


Pressure  Diagram 


FIG.  45.— Theta  and  Pressure  Diagrams  for  7  =  120  and  G  =  0.80. 


From  the  data  in  the  two  preceding  tables,  we  can  obtain  the 
total  ats.  for  each  value  of  the  phase  pressure.  This  is  worked 
through  in  the  following  table: 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       69 


Phase  Pressure. 

Saturation  Ats. 

Ats.  Required  to 
Offset  Armature 
Demagnetization. 

Total  Ats.  for 
120  Amperes  and  0.80 
Power-factor. 

0 

1000 

7200 

8200 

2500 

5  100 

6250 

11  350 

5000 

9400 

5600 

15000 

7200 

20000 

5300 

25300 

CALCULATION  OF  SATURATION  CURVE  FOR  ZERO  POWER 
FACTOR  AND   120  AMPERES 

For  120  amperes  at  zero  power-factor,  the  calculations  are  so 
simple  that  no  diagrams  need  be  drawn.  The  internal  pressures 
are: 

472  volts  for  a  phase  pressure  of . .         0  volts 

(466+2500  =  )2966  volts  for  a  phase  pressure  of  2500  ' ' 
(466+5000  =  )5466  volts  for  a  phase  pressure  of  5000  ' ' 
(466 +7200  =  ) 7666  volts  for  a  phase  pressure  of  7200  " 

Theta  is  Equal  to  90°  for  Loads  of  Zero  Power-factor.     The 

angle  0,  by  which  the  conductors  have  passed  beyond  mid-pole- 
face  position  when  the  instant  arrives  at  which  they  are  carrying 
the  crest  current,  is  substantially  90°  for  all  four  cases,  and 
consequently  the  ats.  required  to  offset  armature  demagnetiza- 
tion remain  constant  at  7200.  The  calculations  are  completed 
in  the  following  table: 


Phase  Pressure. 

Internal 
Pressure. 

Saturation  Ats. 

Ats.  Required'to 
Offset  Armature 
Demagneti- 
zation. 

Total  Ats.  for  120 
Amperes  for  Zero 
Power-factor. 

0 

472 

1000 

7200 

8200 

2500 

2966 

5100 

7200 

12300 

5000 

5466 

9900 

7200 

17100 

7200 

7666 

23000 

7200 

30200 

In  Fig.  46  are  given  saturation  curves  for  loads  of  120  amperes 
at  1.00,  0.90,  0.80  and  0  power-factors. 

In  Fig.  47  the  same  results  are  thrown  into  a  set  of  curves, 
each  relating  to  a  particular  phase-pressure,  the  ordinates  repre- 
senting excitation  and  the  abscissae  representing  power-factor. 


70         POLYPHASE  GENERATORS  AND  MOTORS 


,  Phase  Pressure  in  Volts 

,^-* 

1^=* 
-^ 

—  •• 
^=5 

•  — 
^z 

~  — 

—  — 

,»  •" 

3— 

/ 

x 

x 

Xx 

x^ 

^  —  " 

( 

/ 

/; 

? 

/ 

/ 

/ 

7 

/ 

^/ 

^ 

J 

o/ 

2 

^/ 

'ft 

1 

I/ 

v/ 

u 

^ 

// 

/ 

// 

/ 

fl  / 

/J 

fra 

7 

m 

r 

IN/ 

w 

W\ 

B 

8     5     I     1     g     §     £     f      «j      £     gj     ^     g     go"      o"      gj    3J 

mmf  per  Field  Spool,  in  ats 

FIG.  46. — Saturation  Curves  for  7  =  120  at  Various  Power-factors. 

32,000 

30,000 

28,000 

26,000 

..24,000 

»  22,000 

^  20,000 

g  18,000 

§1*000 

S  14,000 

1 12,000 

1 10,000 

W  8,000 

6,000 

4,000 

2,000 

0.1  0.2  0.3  0.4  0.5  0.6  0.7  0.8  0.9   1.0 
Power  Factor 

FIG.  47. — Curves  showing  the  Required  Excitation  for  7  =  120  at  Various 
Phase  Pressures  and  Power-factors. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       71 

THE  EXCITATION  REGULATION 

The  inherent  regulation  is  not  the  only  kind  of  regulation 
which  it  is  necessary  to  take  into  consideration  in  connection 
with  the  performance  of  a  generator.  There  is  also  the 
"  excitation  regulation."  This,  for  a  given  power-factor  may 
be  defined  as  the  percentage  increase  in  excitation  which  is 
required  in  order  to  maintain  constant  pressure  when  the  out- 
put is  increased  from  no-load  to  any  particular  specified  value  of 
the  current. 

For  our  design  we  have  estimated  that  for  a  phase  pressure 
of  6950  volts  at  no  load,  the  required  excitation  is  : 

14  400  ats. 

For  this  same  phase  pressure  but  with  an  output  of  120  amperes 
per  phase,  the  required  excitations  are  : 

17  140  ats.  for  G  =  1.00 

21  150  ats.  for  G  =  0.90 

22  600  ats.  for  G  =  0.80 
25000  ats.  for  G  =  0 

The  corresponding  values  of  the  excitation  regulation  are: 

=    19-°  per  Cent  f°r  G  =  IW 


oi  i  KH  _  14400 


KH  _  14400  \ 

14  400        X  100=  )46'8  Per  cent  for  0  =  0.90 


(25  000-14 

Excitation  Regulation  Curves.  Curves  plotted  for  given 
values  of  G,  and  of  the  phase  pressure,  with  excitation  as  ordinates 
and  with  current  output  per  phase  as  abscissae,  are  termed 
excitation  regulation  curves.  We  have  values  for  such  curves 
so  far  as  relates  to  7  =  0  and  7  =  120,  but  with  respect  to  higher 
values,  we  have  but  one  point,  namely  : 

7  =  240 
Phase  pressure  =  6950 

G  =  1.00 
Excitation  =  23  000. 


72 


POLYPHASE  GENERATORS  AND  MOTORS 


Let  us  work  out  corresponding  values  for  7  =  240  and  with 
the  other  power-factors,  namely, 

G  =  0.90,  G  =  0.80  and  0  =  0. 


OS68 


SUIBJSBJQ 


For  these  power-factors  and  also  for  G=1.00,  the  theta  and 
pressure  diagrams  are  drawn  in  Fig.  48.  With  the  values  obtained 
from  these  diagrams  the  estimates  may  be  completed  as  follows : 


CALCULATIONS  FOR  2500-KVA.  GENERATOR        73 


Ats.  for 

G. 

e 

Sin  6- 

Offsetting 
Armature 

Internal 
Phase 

Saturation 
Ats 

Total 

Ats 

Demagneti- 

Pressure. 

zation. 

1.00 

28.9° 

0.483 

7000 

7130 

15600 

22730 

0.90 

49.2° 

0.758 

10900 

7490 

18900 

26390 

0.80 

54.9° 

0.817 

11800 

7650 

22200 

29850 

0 

90° 

1.00 

14400 

7880 

35000 

43  000 

The  values  in  the  last  column  and  corresponding  values  already 
obtained  for  7  =  120  and  for  7  =  0,  are  brought  together  in  the 
following  table: 


Excitation  for  6950  Volts  and: 

rt 

1=0. 

7  =  120. 

I  =240. 

1.00 

14  400  ats. 

17  140  ats. 

22  730  ats. 

0.90 

14400  " 

21  150  " 

26390  " 

0.80 

14400  '•' 

22600   " 

29850  " 

0 

14400  " 

25000   " 

43000  " 

These  values  are  plotted  in  the  excitation  regulation  curves 
of  Fig.  49. 


42,000 
40,000 
38,000 
36,000 
34,000 
32,000 
30,000 
28,000 
26,000 
24,000 
22,000 
20,000 
18,000 
16,000 
14,000' 
12,000 
10,000 

5 

/ 

/ 

/ 

/ 

/ 

/ 

/ 

^ 

/ 

f 

* 

^1 

^ 

'•. 

/ 

? 

/ 

& 

£ 

^ 

^ 

/ 

/ 

\S 

^ 

; 

/ 

/ 

rr 

^ 

o^ 

^ 

' 

/, 

^ 

^ 

^. 

^ 

,  —  ' 

^ 

^^ 

~-~* 

< 

^ 

r 

f 
— 

*~~~ 

20    40    60    80  100  120  140  160  180  200  220  240 
Amperes  per  Phase 

FIG.  49. — Excitation  Regulation  Curves  for  6950  Volts. 


74         POLYPHASE  GENERATORS  AND  MOTORS 


Saturation  Curves  for  240  Amperes.  In  the  course  of  the 
previous  investigation  we  have  had  occasion  to  obtain  the  excita- 
tion required  for  a  phase  pressure  of  6950  volts  and  with  an  output 
of  240  amperes.  These  values  are: 


G 

Excitation. 

1.00 

22  730  ats. 

0.90 

26390  " 

0.80 

29850  " 

0 

43000   " 

It  is  not  necessary  to  indicate  the  steps  in  working  out  cor- 
responding values  for  5000  and  2500  volts,  and  it  will  suffice  to 
state  simply  that  the  values  are  those  set  forth  below : 


Excitation  for  Phase  Pressures  of: 

Q 

0  Volts. 

2500  Volts. 

5000  Volts. 

6950  Volts. 

1.00 

16  300  ats. 

17  000  ats. 

18  200  ats. 

22  730  ats. 

0.90 

16300   " 

19000   " 

21  700  '  ' 

26390   " 

0.80 

16300   " 

19500   " 

22700  " 

29850   " 

0 

16300  " 

20600   " 

25400   " 

43000  " 

These  240-ampere  saturation  curves  are  plotted  in  Fig.  50. 

7,000 


6,000 


« 5,000 

(3 


£3,000 
*  2,000 


1,000 


rf      ^      «>'      cd      ^     y       3      jo"      of       o      gj      ,J      ^      oo"      g      g|    ^ 

Excitation  in  ats  per  Field  Spool 

FIG.  50. — Saturation  Curves  for  Various  Values  of  the  Power-factor  and  for 

7=240. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       75 


VOLT-AMPERE  CURVES 

From  the  data  in  Figs.  23,  46  and  50,  relating  respectively 
to  saturation  curves  for  7  =  0,  7  =  120  and  7  =  240,  we  can 
construct  curves  which  may  be  designated  "  volt-ampere " 


8000 


7000 


.2  5000 

I 

c 

1 

g  4000 


3000 


1000 


\ 


\ 


\ 


20       40       60       80      100     120     140     160     180     200     220     240    260 
Current^per  Phase  (in  Amperes) 

FIG.  51. — Volt-ampere   Curves  for  Various  Power-factors  and  for  a  mmf.  of 
17  140  ats.  per  Field  Spool. 

curves,  since  they  are  plotted  with  the  phase  pressure  in 
volts  as  ordinates  and  with  the  current  per  phase,  in  amperes, 
as  abscissae. 

For  any  particular  volt-ampere  curve  the  excitation  and  the 
power-factor  are  constants.  For  the  volt-ampere  curves  in  Fig. 
51,  the  excitation  is  maintained  constant  at  17  140  ats.,  the  mmf. 
required  at  6950  volts,  120  amperes  and  unity  power-factor. 


76         POLYPHASE  GENERATORS  AND  MOTORS 

Comment  is  required  on  the  matter  of  the  value  at  which  the 
volt-ampere  curves  cut  the  axis  of  abscissae.  This  is  seen  to  be 
at  254  amperes.  For  this  current,  the  mmf.  required  to  over- 
come armature  demagnetization  is  obviously: 

254 

— X  7200  =  15  300  ats. 
120 

Since  the  excitation  is  maintained  constant  at  17  140  ats.,  there  is 
a  residue  of : 

17  140 -15  300  =  1840  ats., 

and  this  suffices  to  provide  the  flux  corresponding  chiefly  to  the 
reactance  of  the  end  connections.  We  have  seen  on  pp.  51  and  53 
that  the  reactance  of  the  end  connections  amounts  to : 

(16.3XP-238  =  )3.88  ohms. 
Consequently  for  254  amperes  the  reactance  voltage  is: 

254X3.88  =  990  volts. 
The  IR  drop  is: 

254X0.685  =  174  volts. 

The  impedence  voltage  on  short-circuit  with  250  amperes  is 
consequently : 

\/9902+1742  =  1000  volts. 

From  the  no-load  saturation  curve  of  Fig.  23,  we  see  that 
1800  ats.  are  required  for  a  phase  pressure  of  1000  volts. 

THE  SHORT-CIRCUIT  CURVE 

When  the  stator  windings  are  closed  on  themselves  with  no 
external  resistance,  then  the  field  excitation  required  to  occasion 
a  given  current  in  the  armature  windings  must  exceed  the 
armature  mmf.  by  an  amount  sufficient  to  supply  a  flux  corre- 
sponding to  the  impedance  drop. 

The  impedance  is  made  up  of  two  parts : 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       77 

THE   REACTANCE   OF  THE  END   CONNECTIONS 

AND 
THE   RESISTANCE   OF  THE  WINDINGS 

We  have  seen  (on  p.  51)  that  the  reactance  of  one  phase  at 
25  cycles,  is  16.3  ohms.  Furthermore  we  have  seen  (on  p.  53) 
that  the  reactance  of  the  end  connections  is  23.8  per  cent  of  this 
value,  or: 

0.238X16.3  =  3.88  ohms. 

Also  we  have  seen  (on  p.  22)  that   the   resistance  per  phase,  at 
60°  C.,  is: 

0.665  ohm. 
Consequently,  at  25  cycles,  the  impedance  is : 


V3.882+0.6652  =  3.94  ohms. 

For  any  particular  value  of  the  current,  the  impedance  drop 
is  obtained  by  multiplying  the  current  by  3.94  ohms.  Thus  for 
100  amperes  we  have  an  impedance  drop  of: 

100X3.94  =  394  volts. 

From  the  no-load  saturation  curve  of  Fig.  23,  we  find  that  for 
a  pressure  of  394  volts  per  phase,  a  mmf .  of  700  ats.  per  field  spool 
is  required. 

There  are  25  turns  per  pole  per  phase.  Consequently  for  a 
current  of  100  amperes  per  phase,  the  armature  mmf.  amounts  to: 

2.4X25X100  =  6000  ats. 

Thus  to  send  100  amperes  per  phase  through  the  short-circuited 
stator  windings,  there  is  required  a  mmf.  of: 

700+6000  =  6700  ats.  per  field  spool. 


78         POLYPHASE  GENERATORS  AND  MOTORS 


Making  corresponding  calculations  for  200  and  300  amperes 
we  arrive  at  the  following  results,  which  are  plotted  in  Fig.  52 : 


Current  in 
Armature. 

Impedance 
Drop. 

Saturation 
mmf. 

Armature 
mmf. 

Total  Required 

AtS. 

0  amp. 

0  volts 

Oats. 

Oats. 

Oats. 

100     " 

394     " 

700  " 

6000   " 

6700  " 

200     " 

788     " 

1400  " 

12000   " 

•13400  " 

300     " 

1182     " 

2100  " 

18000  " 

20100  " 

20  000 

i 

18,000 
16,000 
14,000 
12,000 
10,000 
8,000 
6  000 

x 

x 

xl 

x 

x 

X* 

' 

x 

x 

X 

^ 

^ 

4,000 
2.000 

X 

x 

X 

^x 

20      40 


100     120     140     160    180     200     220     240     260     280    300 
Current  per  Phase,  in  Amperes 


FIG.  52. — Short-circuit  Curve  for  2500-kva.  Alternator  with  an  18-mm. 

Air-gap. 

Up  to  300  amperes,  the  short-circuit  characteristic  is  a  straight 
line,  but  at  some  exceedingly-high  current  values  it  may,  in 
machines  designed  with  strong  fields,  bend  upward  quite  a  little 
owing  to  saturation. 

Let  us  now  investigate  the  effect  of  employing  a  lower  mmf. 
in  our  design. 

INFLUENCE    OF   MODIFICATIONS  IN   THE   NO-LOAD 
SATURATION   CURVE 

Let  us  make  the  single  change  of  decreasing  the  radial  depth 
of  the  air-gap  to  6  mm.  in  place  of  the  original  value  of  18  mm. 
The  component  and  resultant  mmf.  at  no  load  will  now  differ  from 


CALCULATIONS  FOR  %500-KVA.  GENERATOR        79 

those    given  in  the  table  on  p.  42  to  the  extent  indicated    in 
the  following  table: 


6000  Volts. 

6950  Volts. 

7500  Volts. 

Air-gap 

3340  ats. 

3850  ats. 

4180  ats. 

Teeth  

90  " 

330  " 

720  " 

Magnet  core 

770  " 

2000   " 

5300   " 

Yoke  
Stator  core  

200  " 

140  " 

320   " 
190   " 

420   " 
230   " 

Total  mmf  

4540  ats. 

6690  ats. 

10  850  ats. 

These  values  are  plotted  in  the  no-load  saturation  curve  of 
Fig.  53. 


8,000 
7,000 
6,000 
5,000 
4,000 
3,000 
2,000 
1,000 


t 


rH        <M        CO       "*        10«0        C-OOOig^ 

Excitation  in  ats  per  Field  Spool 


FIG.  53.  —  No-load  Saturation  Curve  of  2500-kva.  Alternator  with  a  6-mm. 

Air-gap. 


The   calculations   for  the  load  saturation  curves  at  various 
power-factors  are  given  in  the  table  on  p.  80. 


80 


POLYPHASE  GENERATORS  AND  MOTORS 


Phase 
Pressure. 

Internal 
Pressure. 

Saturation 
Ats. 

Ats.  for 
Offsetting 
Armature 
Demagneti- 
zation. 

Total 
Required 
Ats. 

G  =  1.00 
7  =  120 
A=6 

0 
2500 
5000 
6950 
7500 

476 
2630 
5100 
7050 
7590 

350 
1700 
3580 
7200 
12900 

7200 
4350 
2950 
1940 
1800 

7550 
6050 
6530 
9140 
14700 

G=0.90 
7  =  120 
A  =  6 

0 
2500 
5000 
6950 
7200 

476 
2800 
5290 
7250 
7500 

350 
1800 
3700 
8300 
10850 

7200 
5780 
4850 
4450 
4440 

7550 
7580 
8550 
12750 
15290 

G=0.80    [ 
7  =  120 
A  =  6 

0 
2500 
5000 
7200 

476 
2850 
5330 
7560 

350 
1820 
3800 
12000 

7200 
6250 
5600 
5300 

7550 
8070 
9400 
17300 

0=0          f 
7  =  120      ] 
A=6 

0 
2500 
5000 
7200 

476 
2966 
5466 
7666 

350 
1900 
3900 
15500 

7200 
7200 
7200 
7200 

7550 
9100 
11100 
22700 

All  these  saturation  curves  are  brought  together  in  the  curves 
in  Fig.  54. 

From  Figs.  46  and  54  we  obtain  the  following  data: 


Mmf.  per  Field  Spool  Required  for  120  Amp. 
and  a  Phase  Pressure  of  6950  Volts. 

G. 

For  A  =18. 

For  A  =6. 

1.00 

17100 

9140 

0.90 

21000 

12750 

0.80 

22600 

14400 

0 

25000 

17800 

At  first  thought  it  would  appear  to  be  an  advantage  to  obtain 
the  required  full-load  conditions  with  the  much  smaller  amount 
of  field  copper  which  would  suffice  with  the  6-mm.  air-gap. 
Furthermore  the  inherent  regulation  at  full  load  is  identical 
with  that  for  the  18-mm.  air-gap.  Let  us  test  this  last  assertion 
by  obtaining  from  Figs.  23  and  53  the  no-load  pressures  for  the 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       81 


eight  values  of  the  excitation  given  in  the  preceding  table.     We 
have: 


G. 

No  Load  Pressures  Corresponding  to  the 
Excitations  Required  for  6950  Volts  and 
120  Amperes. 

A=18. 

A  =6. 

1.00 

7350 

7350 

0.90 

7600 

7600 

0.80 

7650 

7650 

0 

7700 

7700 

8000 


7000 


Excitation  in  ats  per  Field  Spool 

FIG.  54. — Load  Saturation  Curves  for  2500-kva.  Alternator  with  6-mm. 

Air-gap. 


82         POLYPHASE  GENERATORS  AND  MOTORS 

This  is  a  consequence  of  the  plan  of  limiting  the  pressure  rise 
by  the  saturation  of  the  magnetic  circuit.  The  inherent  regula- 
tion for  both  air-gaps,  has,  for  full-load  current  of  120  amperes, 
the  values  given  in  the  following  table: 

7350  _  6950 
G  =  1  .00       Inherent  regulation  =  —        -»^  -  X  100  =  5.8  per  cent. 


0,0 


If  there  are  objections  to  employing  the  smaller  air-gap,  we 
must  look  elsewhere  for  them.  Let  us  examine  for  instance,  into 
the  question  of  the  amount  of  current  which,  with  normal  excita- 
tion of  9140  ats.  (corresponding  to  6950  volts  and'  120  amperes 
at  unity  power-factor)  could  flow  on  short-circuit.  Without 
going  into  the  matter  of  the  precise  determination  of  the  satura- 
tion mmf.,  let  us  assign  to  this  quantity  the  reasonable  value  of 
700  ats.  This  leaves: 

9140  -700  =  8440  ats. 

for  offsetting  armature  demagnetization.     We  have  seen  that  the 
resultant  armature  mmf.  is  2.4  times  the  mmf.  per  pole  per  phase. 
Thus  we  have: 

8440 
mmf.  per  pole  per  phase  =  -^T  =3500  ats. 

Since  we  have  25  turns  per  pole  per  phase,  the  short-circuit 
current  is  only: 

3500     ,.A 
-Q^-  =  140  amperes. 
Zo 

Thus  with  an  excitation  of  9140  ats.  (the  value  corresponding 
to  full  load  at  unity  power-factor),  a  current  overload  of  only: 


16.7  per  cent 

suffices  to  pull  the  terminal  pressure  down  to  0  volts. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       83 


Similarly  with  an  excitation  of  14  400  ats.,  (the  value  for  120 
amperes  at  6950  volts  and  0.80  power-factor  with  a  6-mm.  air- 
gap),  the  short-circuit  current  is  of  the  order  of: 

14400-1000     00, 
-23X25-  =224  amperes. 

Even  with  this  greater  excitation,  the  current  on  short-circuit 
is  less  than  twice  full-load  current. 

While  the  modern  tendency  is  toward  designing  with  limited 
overload  capacity,  it  is  nevertheless  impracticable  to  employ 


80001 


7000 


6000 


>5000 
.S 

§4000 


3000 


2000 


1000 


'Excitation  Coutant  at  14.400  ats, 
9,140  ats. 


\ 


l\ 


20  40  60  80  100  120  140  160  180  200  220  240 

Current  per  Phase,  in  Amperes. 
FIG.  55. — Volt-ampere  Curves  for  2500-kva.  Alternator  with  6-mm.  Air-gap. 

generators  whose  volt-ampere  characteristics  turn  down  nearly 
so  abruptly  as  do  those  in  Fig.  55  which  represent  these  two 
values.  In  Fig.  55,  the  right-hand  portions  of  the  curves  have 
been  drawn  dotted,  as  it  has  not  been  deemed  worth  while  to  carry 
through  the  calculations  necessary  for  their  precise  predetermina- 
tion. > 

Let  us  now  revert  to  our  original  design  with  the  18-mm. 
air-gap  which  we  have  shown  to  possess  the  more  appropriate 
attributes. 


84         POLYPHASE  GENERATORS  AND  MOTORS 


THE   DESIGN    OF   THE   FIELD    SPOOLS 

Our  generator's  normal  rating  is  2500  kva.  at  a  power-factor 
of  0.90  and  a  phase  pressure  of  6950  volts.  The  current  per 
phase  is  then  120  amperes.  For  these  conditions  the  required 
excitation  is: 

21 150  ats.  per  field  spool. 

The  field  spools  must  be  so  designed  as  to  provide  this  mmf . 
with  an  ultimate  temperature  rise  of  preferably  not  more  than 
45°  Cent,  above  the  temperature  of  the  surrounding  air. 

The  question  of  the  preferable  pressure  to  employ  for  exciting 
the  field,  is  one  which  can  only  be  decided  by  a  careful  considera- 
tion of  the  conditions  in  each  case.  The  pressure  employed  in 
the  electricity  supply  station  for  lighting  and  other  miscellaneous 
purposes,  is  usually  appropriate,  although  it  is  by  no  means  out 
of  the  question  that  it  may  be  good  policy  in  some  cases  to  provide 
special  generators  to  serve  exclusively  as  exciters.  These  exciters 
should,  however,  be  independently  driven.  In  other  words, 
their  speed  should  be  independent  of  the  speed  of  the  generator 
for  which  the  excitation  is  provided.  It  is  the  worst  conceivable 
arrangement  to  have  the  exciter  driven  from  the  shaft  of  the  alter- 
nator, as  any  change  in  the  speed  will  then  be  accompanied  by 
a  more  than  proportional  change  in  the  excitation. 

In  general,  the  larger  the  generator  or  the  more  poles  it  has, 
the  higher  is  the  appropriate  exciting  pressure.  But  it  is  difficult 
to  make  any  statement  of  this  kind  to  which  there  will  not  be 
many  exceptions. 

Let  us  plan  to  excite  our  2500-kva.  generator  from  a  500-volt 
circuit  and  let  us  so  arrange  that  when  the  machine  is  at  its 
ultimate  temperature  of  (20+45  =  )  65°  Cent.,  450  volts  at  the 
slip  rings  shall  correspond  to  an  excitation  of  21  150  ats.  The 
remaining  (500—450  =  )  50  volts  will  be  absorbed  in  the  con- 
trolling rheostat.  It  would  not  be  prudent  to  plan  to  use  up 
the  entire  available  pressure  of  500  volts  when  obtaining  the 
mmf.  of  21  150  ats.,  for  this  would  leave  no  margin  for  discrepan- 
cies between  our  estimates  and  the  results  which  we  should 
actually  obtain  on  the  completed  machine. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       85 
Thus  we  have  450  volts  for  the  eight  spools  in  series,  or: 

450 
r-  =  56.3  volts  per  spool. 

o 

In  Fig.  56  is  shown  a  section  through  the  magnet  core  and  the 
spool. 


— 1149mm-- 


Section  on  A-B  Looking  in  Direction  of  Arrows 
FIG.  56.— Sketches  of  Magnet  Pole  and  Field  Spool  for  2500-kva.  Alternator. 

The  inner  periphery  of  the  spool  is 

26X71+2X60  =  82+120  =  202  cm. 

The  outer  periphery  is  46X^+2X60  =  144+120  =  264  cm. 
For  the  mean  length  of  turn  we  have : 

202+264 
mlt.  = — =  233  cm. 


86         POLYPHASE  GENERATORS  AND  MOTORS 

Suppose  we  were  to  provide  our  normal  excitation  of  21  150  ats. 
by  means  of  a  single  turn  carrying  21  150  amperes.  For  this 
excitation  the  pressure  per  spool  is  56.3  volts.  Consequently  the 
pressure  at  the  terminals  of  our  hypothetical  turn  carrying 
21 150  amperes  is  also  the  56.3  volts  allocated  to  each  of  the 
eight  spools.  The  resistance  of  the  turn  must  consequently  be: 

56'3      0.00266  ohm. 


21150 
We  have  mlt.  =233  cm. 

Therefore  since  the  specific  resistance  of  copper  per  centimeter 
cube,  at  65°  Cent.,  is  0.00000204,  we  have: 

233X0.00000204 
Cross-section  of  the  conductor  =  — 

U.UlLJbo 

=  0.179  sq.cm. 

Now  if  we  were  to  provide  the  entire  excitation  by  a  single 
turn  per  spool  as  above  suggested,  the  loss  in  field  excitation 
would  be: 

500X21150  l 

kw- 


This  is  over  five  times  the  output  of  our  machine.  Consequently 
its  efficiency  would  be  low  —  say  some  18  per  cent.  But  also, 
we  should  be  running  our  conductor  at  a  density  of  : 

21  150 

-f—  7_^-  =  118  000  amperes  per  sq.cm. 
u.  1  1  y 

and  it  would  fuse  long  before  this  density  could  be  reached.  It 
would,  in  fact,  fuse  with  a  current  of  the  order  of  only  some 
1000  amperes.  Also  the  loss  of  10  600  kw.  in  the  field  spools 
would  suffice,  even  if  the  heat  could  be  uniformly  distributed 
through  the  whole  mass  of  the  machine,  to  raise  it  to  an  exceed- 
ingly high  temperature.  Even  a  very  spacious  engine  room 
would  be  unendurable  with  so  great  a  dissipation  of  energy  taking 
place  within  it. 


CALC  ULA  TIONS  FOR  2500-K  VA .  GENERA  TOR       87 

So  let  us  look  into  the  merits  of  employing  10  turns  per  spool 
instead  of  only  one  turn  per  spool. 

Since  we  require  an  excitation  of  21  150  ats.  per  spool,  the 

21  150 
current  will,  in  this  case,  be  only  — — —  =  2115  amperes. 

Since  the  mlt.  is  equal  to  233  cm.,  the  10  turns  will  have  a 
length  of 

10X233  =  2330  cm. 
The  resistance  of  the  10  turns  must  now  be: 


fff -0.0266  ohm. 


Then  we  have : 


Q     ,.         2330X0.00000204     ni7n 
Section  =  -  =  0.179  sq.cm. 


This  is  the  same  value  as  before.  In  fact,  for,  firstly,  a  given 
pressure  at  the  terminals  of  a  spool;  secondly,  a  given  excitation 
to  be  provided,  and  thirdly,  a  given  mlt.,  the  cross-section  of  the 
conductor  is  independent  of  the  number  of  turns  employed  to 
provide  that  excitation,  and  it  is  convenient  to  determine  upon 
the  cross-section  by  first  assuming  that  a  single  turn  will  be 
employed.  Obviously  the  greater  the  number  of  turns  per 
spool,  the  less  will  be  the  current,  and  since  the  terminal  pres- 
sure is  fixed,  the  less  also  will  be  the  power.  Thus  with  our 
second  assumption  of  a  10-turn  coil,  and  only  2115  amperes  in 
the  exciting  circuit,  the  excitation  loss  is  reduced  to: 

500X2115 

-  =1060kw- 


and  the  efficiency  of  our  2500-kva.  machine  would  rise  to  over 
65  per  cent. 

With  the  endeavor  to  obtain  a  reasonably  low  excitation 
loss,  it  is  obviously  desirable  to  employ  as  many  turns  as  we  can 
arrange  in  the  space  at  our  disposal.  We  have  already  seen  that 
this  space  provides  a  cross-section  of  10X20  =  200  sq.cm.  Our 
conductor  has  a  cross-section  of  17.9  sq.  mm.  and  should  thus 


88         POLYPHASE  GENERATORS  AND  MOTORS 


have  a  bare  diameter  of  4.77  mm.  The  curves  in  Fig.  57  give  the 
thicknesses  of  the  insulation  on  single,  double  and  triple  cptton- 
covered  wires  of  various  diameters. 

If,  in  this  case,  we  employ  a  double  cotton  covering,  the  insu- 
lated diameter    will   be    4.77 +(2X0. 18)  =5.13  mm.     The   term 


0.3 


I  0.2 


0.1 


4P 


w 


^ 


a 


2.0 


4.0  6.0  8.0 

Diameter  of  Bare  Conductor,  in  mm 


10.0 


12.0 


FIG.  57. — Thicknesses  of  Insulation  on  Cotton-covered  Wires. 

"  space  factor  "  as  applied  to  field  spools,  is  employed  to  denote 
the  ratio  of  the  total  cross-section  of  copper  in  the  spool,  to 
the  gross  area  of  cross-section  of  the  winding  space.  Attainable 
values  for  the  "  space  factor  "  of  spools  wound  with  wires  of 
various  sizes  and  with  various  insulations,  are  given  in  the 
curves  in  Fig.  58. 

For  the  case  we  are  considering,  we  ascertain  from  the  curves 
that  the  space-factor  may  be  equal  to  0.55.  That  is  to  say:    55 


CALCULATIONS  FOR  2500-KVA.  GENERATOR   89 

per  cent  of  the  cross-section  of  the  winding  space  will  be  copper 
and  the  remaining  45  per  cent  will  be  made  up  of  the  insulation 
and  the  waste  space.  Thus  the  aggregate  cross-section  of  copper 
will  be  200X0.55  =  110  sq.cm.  Consequently  the  number  of 
turns  is: 


110 
0.179 


=  615. 


0.5 


0.4 


0.2 


0.1 


1234  5678 

Bare  Diameter  of  Wire  in  mm 

FIG.  58. — Curves  showing  "Space  Factors"  of  Field  Spools  Wound  with 
Wires  of  Various  Diameters. 


For  the  normal  magnetomotive   force   of  21  150  ats.  per  spool, 
the  exciting  current  is: 


21  150 
615 


=  34.4  amperes. 


The  total  excitation  loss  is: 

500X34.4  =  17200  watts. 


90         POLYPHASE  GENERATORS  AND  MOTORS 
But  of  this  loss  of  17  200  watts: 

crj 

^rX  17  200  =  1720  watts 
ouU 

are  dissipated  in  the  field  regulating  rheostat,  and  only  : 
X  17  200  =  15  500  watts 


are  dissipated  in  the  field  spools.     The  loss  per  spool  is  thus: 
15500 


o 


1940  watts. 


The  next  step  is  to  ascertain  whether  this  will  consist  with  a 
suitably-low  temperature  rise.  The  peripheral  speed  of  our 
rotating  field  is: 

375 

-^-  =  35.0  meters  per  second. 


At  this  speed  and  with  this  very  open  general  construction  with 
salient  poles,  it  will  be  practicable  to  restrict  the  temperature 
rise  to  some  1.3°  rise  per  watt  per  sq.dm.  of  external  cylindrical 
radiating  surface  of  the  field  spool.  We  have: 

External  periphery  of  the  spool  =  26.4  dm. 

Length  of  spool  =  2.0  dm. 

External  cylindrical  radiating  surface  =  26.4  X  2.0  =  52.8  sq.dm. 


Watts  per  sq.dm.  =          =  36.8. 

OZ.o 

Ultimate  temperature  rise  =  36.8X1.  3  =  48°  Cent. 

This  is  a  high  value  and  would  not  be  in  accordance  with  the 
terms  of  usual  specifications.  But  the  modern  tendency  is  to 
take  advantage  of  the  increasing  knowledge  of  the  properties 
of  insulating  materials  and  to  permit  higher  temperatures  in 
low-pressure  windings,  provided  offsetting  advantages  are  thereby 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       91 

obtained.  This  is  the  case  in  moderate-speed  and  high-speed 
polyphase  generators.  The  entire  design  profits  in  great  measure 
by  compressing  the  rotor  into  the  smallest  reasonable  compass. 
In  designing  on  such  lines,  the  space  available  for  the  field  spools 
is  necessarily  restricted.  In  the  case  of  the  design  under  con- 
sideration, we  could,  by  decreasing  the  air-gap,  decrease  the 
required  excitation  and  consequently  also  the  temperature  rise 
of  the  field  spools.  But  we  have  already  seen  that  the  char- 
acteristics of  the  machine  would  be  impaired  by  doing  this. 

For  a  power-factor  of  1.00  the  mmf.  required  per  field  spool 
is  only  17  200  ats.  The  loss  in  the  field  spool  decreases  as  the 
square  of  the  mmf.  Consequently  were  the  machine  required 
for  an  output  of  2500  kva.  at  exclusively  unity  power-factor,  the 
temperature  rise  would  be  only; 


1720Q\2 
,21  1507 


X 48°  =  32°  Cent. 


Thus  for  an  putput  of  2500  kilowatts  at  unity  power-factor, 
the  field  spools  are  actually  33  per  cent  cooler  than  for  an  output 
of  only  2250  kilowatts  at  a  power-factor  of  0.90  (i.e.,  for  an  out- 
put of  2500  kva.  and  a  power-factor  of  0.90). 

If  on  the  contrary,  the  machine  were  required  for  an  output 
of  only  2000  kilowatts  but  at  a  power-factor  of  0.80  (again  2500 
kilovolt-amperes),  the  temperature  rise  of  the  field  spools  would  be: 


It  is  impressive  to  note  that  although  this  output  (2000  kilo- 
watts at  0.80  power-factor),  is  20  per  cent  less  than  an  output 
of  2500  kilowatts  at  unity  power-factor,  the  temperature  rise 
of  the  field  spools  is: 

K4  _  00 

^f^X  100  =  69  per  cent 


greater.  The  losses  and  temperature  rise  in  the  other  parts  of 
the  machine  will  be  the  same  for  both  these  conditions,  since  the 
kilovolt-ampere  output  of  the  machine  is  2500,  in  both  cases. 


92 


POLYPHASE  GENERATORS  AND  MOTORS 


It  is  the  object  of  this  treatise  to  explain  methods  of  design,  and 
consequently  no  particular  advantage  would  be  gained  by  carry- 
ing through  modified  calculations  for  the  purpose  of  providing 
field  spools  which  would  permit  of  carrying  the  rated  load  at 
0.90  power-factor  with  a  temperature  rise  of  only  45°  Cent,  instead 
of  a  temperature  rise  of  48°  Cent.  An  inspection  of  Figs.  21  and  56 
shows  that  there  is  room  for  more  spool  copper  should  its  use 
appear  desirable.  It  would  also  be  practicable  to  decrease  the 
internal  diameter  of  the  magnet  yoke  from  the  present  928  mm. 
down  to — say — 878  mm.  and  increase  the  radial  length  of  the 
magnet  core  (and  consequently  also  the  length  of  the  winding 
space),  by  25  mm, 

THE    CORE    LOSS 

In  polyphase  generators,  the  core  losses  may  be  roughly  pre- 
determined from  the  data  given  in  Table  10. 

TABLE  10.     DATA  FOR  ESTIMATING  THE  CORE  Loss   IN  POLYPHASE 
GENERATORS. 


Density  in  Stator 
Core  in  Lines  per 
Square  Centimeter. 

Core  Loss  in  Stator  Core,  in  kw.  per  (Metric)  Ton  for  Various 
Periodicities. 

^=15. 

~=25. 

^=50. 

6000 

1.1 

2.2 

5.0 

8000 

1.7 

3.0 

7.4 

10000 

2.2 

4.0 

10.0 

12000 

2.6 

5.2 

14000 

3.2 

6.2 

In  the  case  of  the  2500-kva.,  25-cycle  generator  which  we  have 
taken  for  our  example,  the  density  in  the  stator  core  is  10  000, 
lines  per  square  centimeter.  The  core  loss  will  thus  be  on  the 
basis  of  some  4.0  kw.  per  (metric)  ton.  We  must  now  estimate 
the  weight  of  the  stator  core: 

Estimation  of  Weight  of  Stator  Core. 


External  diameter  of  stator  core  =  230  cm. 
Internal  diameter  of  stator  core  =  178  cm. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR   93 

Area  of  the  surface  of  an  annular  ring  with  the  above  external 
and  internal  diameters  is  equal  to: 

|(2302  - 1782)  =  16  700  sq.cm. 

Area  of  the  120  slots  =  120X5.72X2.41  =  1650  sq.cm. 
Net  area  of  surface  of  stator  core  plate  =  16  700  — 1650 

=  15  050  sq.cm. 
In  =  79.2  cm. 

79  2X15  050 
Volume  of  sheet  steel  in  stator  core  =     '  =1.19  cu.m. 

1  UUU  UUU 

Weight  of  1  cu.m.  of  sheet  iron  =  7.8  (metric)  tons. 
Weight  of  stator  core  =  7.8Xl.  19  =  9.3  (metric)  tons. 

On  the  basis  of  a  loss  of  4.0  kw.  per  ton,  we  have: 
Core  loss  =  4.0X9.3  =  37.2  kw. 

Friction  Loss.  No  simple  rules  can  be  given  for  estimating 
the  windage  and  bearing  friction  loss.  The  ability  to  form  some 
rough  idea  of  the  former  can,  in  a  design  of  this  type,  only  be 
acquired  by  long  experience.  It  must  suffice  to  state  that,  for 
the  present  design,  a  reasonable  value  is: 

Windage  and  bearing  friction  loss  =  20  kw. 

Excitation  Loss.  The  loss  in  the  exciting  circuit  is  made  up 
(seep.  90)  of: 

Loss  in  regulating  rheostat  =  1.7  kw. 
Loss  in  field  spools  =  15. 5  kw. 

Stator  I2R  Loss.  The  resistance  of  the  stator  winding  (see 
p.  22)  is: 

0.665  ohm  per  phase. 


94        POLYPHASE  GENERATORS  AND  MOTORS 
and  consequently  we  have: 

Stator  PR  loss  at  rated  load  =  3-X  121°^°'665  =  28.6  kw. 

1UUU 

Total  Loss.     The  total  loss  at  full  load  is  the  sum  of  these 
various  losses: 

I.  Stator  72#  =  28  600  watts 

II.  Field  spool  I2R  =  15500     " 

III.  Field  rheostat  PR  =        1700     " 

IV.  Core  loss  =  37200     " 
V.  Friction  loss  =  20  OOP     " 

Total  loss  at  full  load="  103000  '  ' 
Output  at  full  load  =2250000  '  ' 
Input  at  full  load  =  2  353  000  '  ' 

2250 
Full  -load  efficiency  =  ^^X  100  =  95.6  per  cent. 


CONSTANT   AND   VARIABLE   LOSSES 

Of  the  five  component  losses,  the  last  four  remain  fairty  con- 
stant at  all  loads,  whereas  the  first  loss  (the  stator  PR  loss) 
varies  as  the  square  of  the  load.  It  is  true  that  the  sum  of  the 
second  and  third  losses  decreases  slightly  with  decreasing  load, 
but  in  the  present  machine  the  total  decrease  is  only  in  the  ratio 
of  the  mmf.  at  full  load  and  no  load.  The  twommf.are: 

21  150  ats.  at  rated  load. 
14  400  ats.  at  no  load. 

The  corresponding  values  of  the  total  excitation  loss  are  : 

17  200  watts  at  full  load  (and  6950  volts  per  phase) 
and 


200  -1  70°  watts  at  no  load  <and  695°  volts  Per 
phase) 


CALCULATIONS  FOR  2500-KVA.  GENERATOR        95 
Thus  the  sum  of  the  last  four  losses  decreases  from: 

17  200+37  200+20  000  =  74  400  watts  at  full  load 
down  to 

11  700+37  200+20  000  =  68  900  watts  at  no  load. 
This  decrease  only  amounts  to 
74400-68900 


74400 


X 100  =  7.4  per  cent. 


Thus,  taken  broadly,  we  may  take  the  last  four  component 
losses  as  making  up  an  aggregate  which  we  may  term  the  "  con- 
stant loss,"  and  in  contradistinction  we  may  term  the  first  com- 
ponent the  "  variable  loss."  In  our  design  we  have: 

Variable  loss  =28  600  watts. 
Constant  loss  =  74  400  watts. 

If  we  ignore  the  7  per  cent  decrease  in  the  constant  loss,  we  may 
readily  obtain  the  efficiencies  at  various  loads.  The  method 
will  be  clear  from  an  inspection  of  the  following  estimates: 


EFFICIENCY  AT  ONE-FOURTH  OF  FULL  LOAD 

Variable  loss  =  0.252  X  28  600  =1  800  watts 

Constant  loss  =   74400      " 


Total  loss  at  one-fourth  of  full  load=   76200      " 
Output  =  0.25  X  2  250  000  =  564  000      " 


Input  =640200  watts 

Kfi4.  nnn 
rj(at  one-fourth  load)  =          ~  =  0.882. 


96         POLYPHASE  GENERATORS  AND  MOTORS 


EFFICIENCY  AT  HALF  LOAD 

Variable  loss  =  0.502  X  28  600  =        71 50  watts 
Constant  loss  =      74400    " 


Total  loss  at  half  load  =      81550    ' 

Output  at  half  load  =  1  125  000    " 


Input  =  1  207  000  watts 

ij(at  half  load) 


EFFICIENCY  AT  50  PEK  CENT  OVERLOAD 

Variable  loss  =  1 .502  X  28  600          =      64  500  watts 
Constant  loss  =      74400      " 


Total  loss  at  50  per  cent  overload  =    138  900      " 
Output  at  50  per  cent  overload     =3  380  000      " 


Input  =  3  519  000  watts 

ooorj 

rj(at  50%  overload)  =         =  0.961 . 


LOAD  CORRESPONDING  TO  MAXIMUM  EFFICIENCY 

When  the  variable  losses  have  increased  until  they  equal  the 
constant  losses,  the  efficiency  will  be  at  its  maximum.  The 
corresponding  load  is : 

400 

X  2250  =  1.61X2250 

=  3640kw. 
The  efficiency  is  then: 


3640+74.4+74.4     3789 


0.962. 


CALCULATIONS  FOR  2500-KVA.  GENERATOR       97 

From  this  point  upward,  the  efficiency  will  decrease. 

This  brief  method  of  estimating  the  efficiencies  at  several 
loads,  gives  slightly  too  low  results  at  low  loads  and  slightly  too 
high  results  at  high  loads.  But  the  errors  are  too  slight  to  be 
of  practical  importance;  in  fact  the  inevitable  errors  in  deter- 
mining the  component  losses  are  of  much  greater  magnitude. 

In  Fig.  59  is  plotted  an  efficiency  curve  for  the  above-cal- 
culated values  which  correspond  to  a  power-factor  of  0.90. 


1000  2000  3000  4000  5000 

Output  in  Kilowatts 

FIG.  59.— Efficiency  Curve  of  2500-kva.  Alternator  for  G=0  90. 


DEPENDENCE    OF    EFFICIENCY    ON     POWER-FACTOR    OF 

LOAD 

Let  us  consider  the  load  to  be  maintained  at  2500  kva.  but 
with  different  power-factors.     The  field  excitation  will  be: 


For  G=  1.00: 

For  G  =  0.90: 
For  G  =  0.80: 


17200 
21  150 


22500 
21  150 


X 17  200  =  14  000  watts. 


17  200  watts. 


X 17  200  =  18  300  watts. 


POLYPHASE  GENERATORS  AND  MOTORS 
The  total  losses  become: 


G 

1.00 
0.90 
0.80 


Total  Losses 
99  800  watts 
103  000      " 
104 100      " 


The  outputs,  inputs  and  efficiencies  become: 


G. 

Output  Corresponding 
to  2500  kva. 

Input. 

•n 

1.00 

2500  kw. 

2600  kw. 

0.962 

0.90 

2250    " 

2353    " 

0.956 

0.80 

2000    " 

2104    " 

0.949 

This  brings  out  the  importance  of  employing  a  considerable 
proportion  of  over-excited  synchronous  motors  to  offset  the  lag- 
ging load  corresponding  to  induction  motors.  This  question  is 
further  discussed  in  Chapter  VI,  entitled  "  Synchronous  Motors 
versus  Inductor  Motors." 


CHAPTER  III 

POLYPHASE  GENERATORS  WITH  DISTRIBUTED  FIELD 
WINDINGS 

THE  type  of  polyphase  generator  with  salient  poles  which 
has  been  described  in  the  last  chapter  has  served  excellently 
as  a  basis  for  carrying  through  a  set  of  typical  calculations. 
Salient-pole  generators  are  chiefly  employed  for  slow-  and  medium- 
speed  ratings.  But  for  the  high  speeds  associated  with  steam- 
turbine-driven  sets,  rotors  with  distributed  field  windings  are 
practically  universally  employed  in  modern  designs.  It  is  not 
proposed  to  carry  through  the  calculations  for  a  design  of  this 
type.  While  there  are  a  good  many  differences  in  detail,  to 
which  the  professional  designer  gives  careful  attention,  the 
underlying  considerations  are  quite  of  the  same  nature  as 
in  the  case  of  salient-pole  designs.  In  Figs.  60  and  61  are 
shown  photographs  of  a  salient-pole  rotor  and  a  rotor  with 
a  distributed  field  winding.  The  former  (Fig.  60),  is  for 
a  medium  speed  (514  r.p.m.),  water-wheel  generator  with  a 
rated  capacity  for  1250  kva.  The  latter  (Fig.  61),  is  for  a  750- 
r.p.m.,  steam-turbine-driven  set,  with  a  rated  capacity  for  15000 
kva.  The  former  has  14  poles  and  the  latter  4  poles.  The 
former  is  for  a  periodicity  of  60  cycles  per  second,  and  the  latter 
for  a  periodicy  of  25  cycles  per  second. 

An  inherent  characteristic  of  high-speed  sets  relates  to  the 
great  percentage  which  the  sum  of  the  core  loss,  windage  and 
excitation  bears  to  the  total  loss.  In  our  2500-kva.  salient-pole 
design  for  375  r.p.m.,  the  core  loss  amounted  to  some  37  000 
watts,  the  windage  and  bearing  friction  to  20  000  watts,  and  the 
excitation  to  17  000  watts,  making  an  aggregate  of  74  000  watts 
for  the  "  constant  "  losses,  out  of  a  total  loss  at  full-load  of 
103  000  watts.  But  in  a  design  for  2500  kva.  at  3600  r.p.m., 
(i.e.,  for  very  nearly  10  times  as  great  a  speed),  the  core  loss, 

99 


100       POLYPHASE  GENERATORS  AND  MOTORS 

bearing-friction,  windage,  and  excitation  would  together  amount 
to  some  72  000  watts  out  of  a  total  of  some  80  000  watts.     A 


FIG.  60.— Salient  Pole  Rotor  for  a  14-pole,  1250-kva.,  60-cycle,  514  r.p.m., 
3-phase  Alternator,  built  by  the  General  Electric  Co.  of  America. 

representative  distribution  of  the  losses  for  a  2500-kva.,  0.90- 
power-f actor,  12  000- volt  polyphase  generator  would  be: 

Armature  PR  loss 8  000  watts 

Excitation  PR  loss 9  000     " 

Core  loss 32000 

Windage  and  bearing  friction  loss  31  000 

Total  loss  at  full  load 80  000     " 

Output  at  full  load 2  250  000 

Input  at  full  load 2  330  000     " 

Efficiency  at  full  load 96.6  per  cent 


WITH  DISTRIBUTED  FIELD  WINDINGS         101 

A  result  of  the  necessarily  large  percentage  which  the 
"constant"  losses  bear  to  the  total  losses,  is  that  the  efficiency 
falls  off  badly  with  decreasing  load,  In  this  instance  we  have: 

Variable  losses  =  8  kw. 
Constant  losses  =  72  kw. 

The  efficiencies  at  various  loads  work  out  as  follows: 

Load.  Efficiency. 

88.5  per  cent 

i 93.9       " 

1.00 .  96.6      " 


FIG.  61. — Rotor  with  Distributed  Field  Winding  for  a  4-pole,  15  000-kva., 
25-cycle,  750  r.p.m.,  3-phase  Alternator,  built  by  the  General  Electric 
Co.  of  America. 


On  pp.  94  to  96  the  efficiencies  of  the  375-r.p.m.  machine  for 
this  same  rating  were  ascertained  to  be: 


Load. 

i 

1.00.  . 


Efficiency. 

88.2  per  cent 

93.1       " 

.  95.6      " 


102       POLYPHASE  GENERATORS  AND  MOTORS 


For    a    100-r.p.m.,     2250-kw.,     0.90-power-f actor,     25-cycle 
design,  the  efficiencies  would  have  been  of  the  following  order: 

Load.  Efficiency. 

J 88.0  per  cent 

i 92.6      " 

1.00 94.6      " 

The  values  may  be  brought  together  for  comparison  as  follows : 
EFFICIENCIES 


Load. 

100  r.p.m. 

375  r.p.m. 

3600  r.p.m. 

a 

88.0 

88.2 

88.5 

i 

92.6 

93.1 

93.9 

1.00 

94.6 

95.6 

96.6 

While  it  is  within  the  designer's  power  to  modify  the  inherent 
tendencies  corresponding  to  the  rated  speeds,  nevertheless  the 
values  above  set  forth  are  representative.  Thus  while  a  full- 
load  efficiency  of  96.6  per  cent  is  obtained  for  the  3600-r.p.m. 
design,  the  full-load  efficiency  of  the  100-r.p.m.  design  is  only 
94.6  per  cent.  At  quarter  load  the  efficiency  is  practically 
as  high  for  the  100-r.p.m.  design  as  for  the  3600-r.p.m.  design. 
In  judging  of  improvements  in  efficiency,  it  is  the  decrease  in 
the  percentage  of  losses  which  should  be  considered.  Thus 
when  the  efficiency  is  increased  from  94.6  per  cent  to  96.6  per 
cent,  the  losses  are  decreased  from  5.4  per  cent  of  the  input  to 
3.4  per  cent  of  the  input.  The  decrease  in  the  losses  is  thus: 


per  Cent< 


In  a  polyphase  generator  driven  by  a  steam  turbine,  it  is 
now  almost  universal  practice  totally  to  enclose  the  generator, 
except  so  far  as  relates  to  the  provision  of  suitable  inlets  and 
outlets  for  the  circulating  air,  which  is  usually  driven  through 
the  machine  by  means  of  fans  located  on  the  rotor.  Consequently, 
in  this  type  of  machine,  a  step  in  the  calculation  relates  to  esti- 
mating the  supply  of  air  required  suitably  to  limit  the  temperature 


WITH  DISTRIBUTED  FIELD  WINDINGS         103 

rise,  and  so    to    proportion    the    passages  as  to  transmit    the 
air  in  the  quantities  thus  ascertained  to  be  necessary. 

In  our  3600-r.p.m.,  2250-kw.  generator,  the  losses  at  full 
load  amount  to  80  kw.  If  the  heat  corresponding  to  this  loss 
is  to  be  carried  away  as  fast  as  it  is  produced,  then  we  must 
circulate  sufficient  air  to  abstract: 

80  kw.-hr.  per  hr. 

A  convenient  starting  point  for  our  calculation  is  from  the 
basis  that: 

1.16  w.-hr.  raises  1  kg.  of  water  1°  Cent. 

The  specific  heat  of  air  is  0.24;  that  is  to  say,  it  requires 
only  0.24  times  as  much  energy  to  raise  1  kg.  of  air  by  1  degree 
Cent,  as  is  required  to  raise  1  kg.  of  water  by  1  degree  Cent. 

Consequently,  to  raise  by  1  degree  Cent.,  the  temperature 
of  1  kg.  of  air,  requires  the  absorption  of  : 

f 

1.16X0.24  =  0.278  w.-hr. 

One  kilogram  of  air  at  atmospheric  pressure  and  at  30  degrees 
Cent,  occupies  a  volume  of  0.85  cu.m.  Therefore,  to  raise  1 
cu.m.  of  air  by  1  degree  Cent,  requires: 


If,  for  the  outgoing  air,  we  assume  a  temperature  25  degrees 
above  that  of  the  ingoing  air,  then  every  cu.m.  of  air  circulated 
through  the  machine  will  carry  away: 

0.327X25  =  8.2  w.-hr. 

We  must  arrange  for  sufficient  air  to  carry  away: 
80  000  w.-hr.  per  hour. 


104       POLYPHASE  GENERATORS  AND  MOTORS 

\ 
Consequently  we  must  supply: 

80000 

8.2 

or: 


=  9800  cu.  m.  per  hour; 


9800     1ft0 

-—-  =  163.  cu.  m.  per  minute. 

In  dealing  with  the  circulation  of  air  it  appears  necessary  to 
make  the  concession  of  employing  other  than  metric  units.  We 
have: 

1  cu.m.  =  35.4  cu.ft. 


Therefore  in  the  case  of  our   2250-kw.  generator,  we   must 
circulate : 

163X35.4  =  5800  cu.ft.  per  min. 


CHAPTER  IV 

THE  DESIGN  OF  A  POLYPHASE  INDUCTION  MOTOR  WITH  A 
SQUIRREL-CAGE  ROTOR 

THE  polyphase  induction  motor  was  brought  to  a  commer- 
cial stage  of  development  about  twenty  years  ago.  Many  tens 
of  thousands  of  such  motors  are  now  built  every  year.  The  design 
of  polyphase  induction  motors  has  been  the  subject  of  many 
elaborate  investigations  and  there  has  been  placed  at  the  disposal 
of  engineers  a  large  number  of  practical  rules  and  data. 

The  design  of  such  a  motor  may  proceed  from  any  one  of 
many  starting  points  and  each  designer  has  his  preferred  method. 
The  author  proposes  to  indicate  the  method  which  he  has  found 
to  be  the  most  useful  for  his  purposes.  It  must  not  be  inferred 
that  any  set  of  rules  can  be  framed  which  will  lead  with  certainty 
to  the  best  design  for  any  particular  case.  The  most  which 
can  be  expected  is  that  the  rules  shall  lead  to  a  rough  preliminary 
design  which  shall  serve  to  fix  ideas  of  the  general  orders  of  dimen- 
sions. Before  he  decides  upon  the  final  design,  the  enterprising 
designer  will  carry  through  a  number  of  alternative  calculations 
in  which  he  will  deviate  in  various  directions  from  the  original 
design.  A  consideration  of  the  several  alternative  results  at 
which  he  will  thus  arrive,  will  gradually  lead  him  to  the  most 
suitable  design  for  the  case  which  he  has  in  hand. 

The  method  of  design  will  be  expounded  in  the  course  of 
working  through  an  illustrative  example. 

ILLUSTRATIVE   EXAMPLE 

Let  it  be  required  that  a  three-phase  squirrel-cage  induction 
motor  be  designed.  The  normal  rating  is  to  be  200  hp.  and  the 
motor  is  to  be  operated  from  a  1000-volt,  25-cycle  circuit.  It 
is  desired  that  its  speed  shall  be  in  the  close  neighborhood  of 
500  r.p.m. 

105 


106       POLYPHASE  GENERATORS  AND  MOTORS 

Determination  of  the  Number  of  Poles.    Denoting  the  speed 

in  revolutions  per  minute  by  R,  then  the  speed  in  revolutions 

-p 

per  second  is  equal  to  ~~.     If  we  denote  the  number  of  poles 

by  P,  and  the  periodicity  in  cycles  per  second  by  ^,  then  we 
have: 


In  our  case  we  have  : 

~  =  25  72  =  500. 

Therefore  : 

2X60X25 
500 

If  we  design  the  motor  with  6  poles,  the  "  synchronous  " 
speed  will  be  500  r.p.m.  At  no-load,  the  motor  runs  at  practi- 
cally its  "  synchronous  "  speed;  that  is  to  say,  its  "  slip  "  is 
practically  zero.  The  term  "  slip  "  is  employed  to  denote  the 
amount  by  which  the  actual  speed  of  the  motor  is  less  than  the 
"  synchronous  "  speed.  An  appropriate  value  for  the  slip  of 
our  motor  at  its  rated  load,  is  some  2  per  cent,  or  even  less. 
Taking  it  for  the  moment  as  2  per  cent,  we  find  that,  at  rated 
load,  the  speed  will  be  500-0.02X500  =  490  r.p.m. 

The  variations  in  the  speed  between  no  load  and  full  load 
are  so  slight  that  at  many  steps  in  the  calculations  the  speed 
may  be  taken  at  the  approximate  value  of  500  r.p.m.,  thus 
avoiding  superfluous  refinements  which  would  merely  complicate 
the  calculations  and  serve  no  useful  purpose. 

Rated  Output  Expressed  in  Watts.  Since  one  horse-power 
is  equal  to  746  watts,  the  rated  output  of  our*200-h.p.  motor 
may  also  be  expressed  as  200X746  =  149  200  watts. 

Determination  of  T,  the  Polar  Pitch.  The  distance  (in  cm.) 
measured  at  the  inner  circumference  of  the  stator,  from  the  center 
of  one  pole  to  the  center  of  the  next  adjacent  pole,  is  termed 
the  "  polar  pitch  "  and  is  denoted  by  the  letter  T.  Rough  pre- 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  107 

liminary  values  for  T  are  given  in  Table  11  for  designs  for  25  and 
50  cycles  for  a  wide  range  of  outputs  and  speeds.  These  values 
have  been  found  by  experience  to  be  appropriate. 


TABLE  11. — PRELIMINARY  VALUES  FOR  T  (THE  POLAR  PITCH)  FOR  USE  IN 
DESIGNING  THREE-PHASE  INDUCTION  MOTORS 


Rated  Output. 

25  Cycles. 

H.P. 

Watts. 

P=4. 
R  =750. 

P=6. 
#=500. 

P=8. 
R  =375. 

P=10. 
R  =300. 

P  =  12. 

R  =250. 

P=14. 

R=218. 

5 

10 
20 

3720 
7460 
14900 

17.0 
19.0 
22.0 

14.0 
16.0 
19.0 

12.0 
14.0 
17.0 

13.0 
15.5 

14.0 

40 
60 
80 

29900 
44700 
59700 

26.0 
28.0 
30.0 

22.0 
24.0 
26.0 

20.0 
22.0 
24.0 

18.0 
20.0 
21.5 

17.0 
19.0 
20.5 

16.0 
18.0 
19.0 

100 
150 
200 

74600 
112000 
149000 

31.5 
35.0 
38.0 

27.0 
30.0 
32.5 

24.5 
27.0 
29.5 

23.0 
25.0 
27.5 

21.5 
23.5 
26.0 

20.0 
22.0 
24.0 

300 
400 
500 

224000 
299000 
373000 

43.0 
47.0 
50.0 

37.0 
40.0 
42.0 

33.0 
36.0 
37.5 

30.5 
33.0 
34.5 

29.0 
31.0 
32.5 

27.0 
29.0 
30.0 

Rated 
Output 

50  Cycles. 

! 
HP. 

Watts 

P  =  4 

72=1500 

P=6 
72=1000 

P  =  8 
72  =  750 

P=10 

R  =  600 

P=12 
72=500 

I 
P=14 
72  =  429 

P=16 
72  =  375 

P=20 
72=300 

P  =  24 
72  =  250 

P  =  28 
72  =  214 

5 
10 
20 

3720 
7460 
14900 

16.0 
17.0 
19.0 

13.0 
14.0 
16.0 

11.0 
12.0 
14.0 

11.0 
13.0 

12.0 

11.5 

40 
60 
80 

29900 
44700 
59700 

22.0 
24.0 
25.5 

19.0 
20.5 
22.0 

17.0 
18.5 
19.5 

15.5 
17.0 
18.0 

14.5 
16.0 
17.0 

13.5 
15.0 
16.0 

13.0 
14.5 
15.5 

13.5 
14.5 

13.5 

100 
150 
200 

74600 
112000 
149000 

27.0 
29.5 
31.5 

23.0 
25.0 
27.0 

20.5 
22.5 
24.5 

19.0 
21.0 
23.0 

18.0 
20.0 
21.5 

17.0 
18.5 
20.5 

16.0 
17.5 
19.5 

15.0 
16.5 
18.0 

14.0 
15.5 
17.0 

13.5 
15.0 
16.9 

300 
400 
500 

224000 
299000 
373000 

35.0 
38.0 
40.0 

30.0 
32.5 
34.5 

27.0 
29.5 
31.5 

25.0 
27.0 
29.0 

24.0 
25.5 
27.0 

22.5 
24.0 
25.5 

21.5 
23.0 
24.0 

20.0 
21.5 
22.5 

18.5 
20.0 
21.5 

17.5 
19.0 
20.5 

For  our  200-h.p.,  6-pole,  25-cycle  design,  we  find  from  Table 
11,  the  value: 


cm. 


108       POLYPHASE  GENERATORS  AND  MOTORS 


THE   OUTPUT   COEFFICIENT 

The  next  step  relates  to  the  determination  of  a  suitable  value 
for  ?,  the  "  Output  Coefficient,"  which  is  defined  by  the  following 
formula : 


w 


in  which 

W  =  Rated  output  in  watts,   (which  is  equal  to  746  times 

the  rated  output  in  h.p.), 
D  =  Diameter   at   air-gap,    in   decimeters,    i.e.,    the   internal 

diameter  of  the  stator, 
Xg  =  Gross  core  length,  in  decimeters, 
R  =  rated  speed,  in  revolutions  per  minute. 

As  in  the  earlier  chapters  of  this  treatise,  the  symbols  D  and 
Xgf  will  sometimes  be  employed  for  denoting  respectively  the 
air-gap  diameter  and  the  gross  core  length,  expressed,  as  above, 
in  decimeters,  but  more  usually  they  will  denote  these  quantities 
as  expressed  in  centimeters.  The  student  can  soon  accustom 
himself  to  distinguishing,  from  the  magnitudes  of  these  quan- 
tities, whether  decimeters  or  centimeters  are  intended,  and 
thus  will  not  experience  any  difficulty  of  consequence,  in  this 
double  use  of  the  same  symbols. 

Values  of  5  suitable  for  preliminary  assumptions  are  given 
in  Table  12.  In  this  table,  £  is  given  as  a  function  of  P  and  T. 

For  our  design  we  have : 

p  =  6,     T  =  32.5  cm. 

The  corresponding  value  of  ?  in  Table  12  is  about  2.0.  But 
let  us  for  our  design  be  satisfied  with  a  less  exacting  value  and 
take: 

=  1.80. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   109 

Therefore : 

1.80.        1492°° 


149  200 


TABLE   12. — PRELIMINARY  VALUES  FOR  £  (THE  OUTPUT  COEFFICIENT)  FOR 
USE  IN  DESIGNING  THREE-PHASE  SQUIRREL-CAGE  INDUCTION  MOTORS. 

(The  figures  at  the  heads  of  the  vertical  columns  give  the  numbers  of  poles.) 


r,  the 
polar 
pitch  (in 
cm.)! 

4. 

6. 

: 

8. 

10. 

12. 

14. 

16. 

18. 

20. 

22. 

24. 

11 

0.80 

1.07 

1.12 

1.15 

1.14 

1.12 

1.11 

1.10 

1.08 

1.05 

12 

0.95 

1.14 

1.19 

1.21 

1.20 

1.20 

1.18 

1.16 

1.14 

1.12 

14 

1.13 

1.28 

1.30 

1.31 

1.30 

1.28 

1.26 

1.26 

1.25 

1.25 

16 

0.85 

1.28 

1.40 

1.41 

1.41 

1.41 

1.40 

1.40 

1.40 

1.40 

1.40 

18 

1.11 

1.40 

1.50 

1.50 

1.51 

1.51 

1.51 

1.51 

1.51 

1.51 

1.51 

20 

1,27 

1.52 

1.59 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

1.60 

22 

1.42 

.63 

1.69 

1.70 

1.70 

1.70 

1.70 

1.70 

1.70 

1.69 

1.69 

24 

1.53 

.72 

1.78 

1.79 

1.80 

1.80 

1.80 

1.80 

1.80 

1.80 

1.80 

26 

1.63 

.82 

1.86 

1.87 

1.87 

1.87 

1.88 

1.88 

1.88 

1.88 

1.88 

28 

1.71 

.90 

1.93 

1.93 

1.94 

1.94 

1.95 

1.95 

1.95 

1.95 

1.95 

30 

1.79 

.97 

1.99 

2.00 

2.00 

2.01 

2.01 

2.01 

2.02 

2.02 

2.02 

35 

1.90 

2.08 

2.10 

2.11 

2.11 

2.11 

2.12 

2.12 

2  12 

2.12 

2.12 

40 

1.98 

2.17 

2.20 

2.20 

2.21 

2.21 

2.22 

45 

2.04 

2.21 

2.26 

2.28 

50 

2.07 

2.23 

2.30 

But  D  (in  crn.),  is  equal  to: 


Therefore : 


6X32.5 


110       POLYPHASE  GENERATORS  AND  MOTORS 
Therefore  :. 

166 


(in  dm.)  = 


r- 


dm.  or  43.0  cm. 


D  and  \g  and  T  are  the  three  characteristic  dimensions  of  the 
design  with  which  we  are  dealing. 

D2\g  (with  D  and  \g  expressed  in  decimeters),  is  also,  in 
itself,  a  useful  value  to  obtain  at  an  early  stage  of  the  calculation 
of  a  design.  We  have,  for  our  motor; 


PRELIMINARY  ESTIMATE  OF  THE  TOTAL  NET  WEIGHT 

A  rough  preliminary  idea  of  the  total  net  weight  of  an  induc- 
tion motor  may  be  obtained  from  a  knowledge  of  its  D2\g. 
The  "  Total  Net  Weight  "  may  be  taken  as  the  weight  exclusive 
of  slide  rails  and  pulley.  In  Table  13,  are  given  rough  repre- 
sentative values  for  the  Total  Net  Weights  of  induction  motors 
with  various  values  of  D2\g. 

TABLE  13. — VALUES  OF  THE  TOTAL  NET  WEIGHT  OF  INDUCTION  MOTORS. 


10 
20 
40 

60 

80 

100 

150 
200 
250 

300 
350 
400 


Total  Net  Weight  in 
Metric  Tons,  (i.e.,  in 
Tons  of  2204  Lbs.). 


0.27 
0.40 
0.73 

0.98 
1.20 
1.40 

1.90 
2.30 
2.70 

3.00 
3.25 
3.45 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   111 


PRELIMINARY  ESTIMATE  OF  THE  TOTAL  WORKS  COST 

The  Total  Works  Cost  will  necessarily  also  be  a  very  indefinite 
quantity,  susceptible  to.  large  variations  with  variations  in  the 
proportions  and  arrangement  of  a  design.  It  is  even  more 
greatly  dependent  upon  the  equipment  and  management  of 
the  Works  at  which  the  motor  is  manufactured.  Neverthless, 
some  rough  indication  appropriate  for  squirrel-cage  induction 
motors  is  afforded  by  the  data  in  Table  14. 

TABLE   14. — TOTAL  WORKS  COST  OF  SQUIRREL-CAGE  INDUCTION  MOTORS. 


Total  Net  weight  of 
Motor  in  Metric  Tons. 


0.20 
0.40 
0.60 

0.80 
1.00 
1.50 

2.00 
2.50 
3.00 

3.50 
4.00 


Total  Works  Cost  per 
Ton,  in  Dollars. 


310 
300 
290 

285 
280 
270 

260 
250 
240 

230 
225 


For  our  200-h.p.  motor,  we  have  D2\g  =  166. 

From  Table  13,  we  ascertain  that  the  Total  Net  Weight 
is  some  2.00  tons.  From  Table  14,  it  is  found  that  the  Total 
Works  Cost  is  of  the  order  of  $260.  per  ton.  Consequently  we 
have  Total  Works  Cost -2.00X260  =  $520. 

ALTERNATIVE    METHOD    OF    ESTIMATING    THE    TOTAL 
WORKS  COST 

An  alternative  method  of  estimating  the  T.W.C.  of  an  induction 
motor  is  based  on  the  following  formula: 


TWC  (in  dollars)  =KxDXfrg+0.7i), 


112       POLYPHASE  GENERATORS  AND  MOTORS 

where  D,  Xg,  and  T  are  given  in  centimeters.     K  is  obtained 
from  Table  15. 

TABLE  15 — VALUES  OF  K  IN  FORMULA  FOR  T.  W.  C. 


Air-gap  Diameter,  D. 
in  Centimeters. 

Values  of  K. 

10 
20 
40 

0.098 
0.105 
0.112 

60 
80 
100 

0.120 
0.128 
0.133 

150 
200 

0.140 
0.148 

In  the  200-h.p.  motor  which  is  serving  us  as  an  example, 
we  have: 

T  =  32.5,         0.7T  =  22.7, 


D  =  62.0,        K  (from  Table  15)  =0.122, 
TWC  =  0.122X62X65.7  =  $508. 


Thus  the  results  by  the  two  methods  are  $520  and  $508 
which  are  in  good  agreement  with  each  other.  Usually  the 
agreement  will  be  far  less  close  and  an  average  of  the  two  values 
is  preferable  as  a  guide.  It  is  interesting  to  note  that  the  TWO 
per  rated  horse-power  is,  in  the  case  of  this  motor,  some: 

514  -*2  57 
200" 

The  motor  could  not,  of  course,  be  bought  at  any  such  price, 
for  the  TWO  merely  covers  all  the  costs  incurred  up  to  the 
delivery  of  the  completed  machine  to  the  shipping  department, 
to  be  packed.  Selling  and  shipping  expenses,  as  also  profits, 
must  be  added  to  the  TWC  to  arrive  at  the  price  at  which  the 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR    113 

machine  should  be  sold  and  it  is  not  unusual  to  find  the  selling 
price  over  double  the  Total  Works  Cost.  This  is  one  of  the 
penalties  of  the  competitive  system  of  supplying  the  needs  of 
mankind. 

The  Peripheral  Speed.  It  is  well,  before  proceeding  further 
with  the  design,  to  calculate  the  peripheral  speed.  Let  us  denote 
by  S,  the  peripheral  speed,  expressed  in  meters  per  second, 


R 
'  100  A60' 


For  our  design : 


0    xX62X500  ' 

100X60    =         meters  Per  second. 


In  this  instance  the  peripheral  speed  is  very  low,  and  does 
not  constitute  a  limiting  consideration  from  the  point  of  view 
of  mechanical  strength.  For  other  speeds,  ratings,  and  peri- 
odicities, the  preliminary  data  as  derived  from  the  rules  which 
have  been  set  forth,  might  lead  to  an  undesirably-high  peripheral 
speed.  Consequently  it  is  well  to  ascertain  the  peripheral  speed 
at  an  early  stage  of  the  calculations  and  arrange  to  reduce  D 
and  T  in  cases  where  the  electrical  design  ought  to  be  sacrificed 
in  some  measure  in  the  interests  of  improving  the  mechanical 
design. 


PERIPHERAL  LOADING 

We  shall  next  deal  with  the  determination  of  the  number  of 
conductors  to  be  employed.  The  product  of  the  number  of 
conductors  and  the  current  per  conductor,  (i.e.,  the  ampere- 
conductors), constitutes  a  quantity  to  which  the  term  "  peripheral 
loading  "  may  be  applied.  Designers  find  from  experience  that 
it  is  desirable  to  employ  certain  definite  ranges  of  values  for  the 
peripheral  loading  per  centimeter  of  periphery,  measured  at  the 
air-gap.  In  Table  16,  are  given  values  which  will  serve  as  pre- 
liminary assumptions  for  a  trial  design. 


114       POLYPHASE  GENERATORS  AND  MOTORS 


TABLE  16. — PRELIMINARY  ASSUMPTIONS  FOR  THE  PERIPHERAL  LOADING  OF 
AN  INDUCTION  MOTOR. 


5  (in  Centimeters). 

Stator  Ampere  Conductors  per  Centi- 
meter of  Gap  Periphery. 

25  Cycles. 

50  Cycles. 

15 
20 
25 

140 
180 
220 

180 
220 

270 

30 
40 
60 

270 
320 
370 

310 
350 
400 

80 
100 
120 

380 
390 
400 

420 
430 
440 

For  motors  wound  for  very  low  pressures,  (say  250  volts), 
somewhat  higher  values  may  be  employed,  especially  in  the 
larger  sizes.  On  the  contrary,  for  motors  wound  for  high  pres- 
sures— say  2500  volts  or  more — it  is  necessary  to  employ  for 
the  ampere  conductors  per  centimeter  of  periphery,  lower  values 
than  those  indicated  in  the  table,  especially  in  motors  of  very 
small  diameter.  Indeed  the  loss  of  space  in  providing  for  slot 
insulation  renders  it  very  undesirable  to  wind  small  motors  for 
very  high  pressures.  It  is  better,  in  such  cases,  to  interpose 
step-down  transformers  between  the  supply  system  and  the 
motor  or  motors. 

Experienced  designers  will  find  occasions  where  the  periphery 
may,  and  should,  be  loaded  with  ampere  conductors  much  more 
highly  than  corresponds  with  the  data  in  Table  16.  On  the 
other  hand,  there  are  often  difficult  ratings,  (as  regards  overload 
capacity  and  other  features  which  will  later  come  in  for  con- 
sideration), where  the  peripheral  loading  should  (and  must)  be 
much  lower  than  the  values  indicated  in  Table  16.  As  a  matter 
of  fact,  the  rated  speed  and  output,  the  periodicity  and  pressure, 
and  also  the  stipulated  instantaneous  overload  capacity,  all  require 
to  be  taken  into  consideration.  But  at  this  early  stage  in  the 
design,  it  is  useful  to  take  a  value  from  Table  16.  For  our  case, 
D  is  equal  to  62,  and  the  corresponding  value  for  the  peripheral 
loading  is  seen  to  be  372  ampere  conductors  per  centimeter. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   115 
Consequently  the  total  peripheral  loading  is: 
62XxX372  =  72  500  ampere-conductors; 


or 


72  500 

— ~- —  =  24  200  ampere-conductors  per  phase. 


From  this  product  we  wish  next  to  segregate  the  ampeers  and 
the  conductors.  We  may  do  this  by  dividing  the  ampere- 
conductors by  the  full-load  current  per  phase.  We  cannot 
estimate  the  precise  value  of  the  full-load  current  per  phase 
until  the  design  has  been  completed,  as  its  precise  value  depends 
upon  the  efficiency  and  power-factor  at  full  load.  But  we  cannot 
complete  the  design  without  determining  upon  the  suitable 
number  of  conductors  to  employ.  Hence  it  becomes  necessary 
to  have  recourse  to  tables  of  rough  approximate  values  for  the 
full-load  efficiency  and  the  power-factor  of  designs  of  various 
ratings.  Such  values  are  given  in  Tables  17  and  18. 

EFFICIENCY  AND  POWER-FACTOR 

For  the  case  of  squirrel-cage  induction  motors  for  moderate 
pressures,  and  in  the  absence  of  any  specially  exacting  require- 
ments as  regards  capacity  for  carrying  large  instantaneous  over- 
loads, we  may  proceed  from  the  basis  of  the  rough  indications 
in  Tables  17  and  18.  From  these  tables  we  obtain: 

Full-load  efficiency  =  91  per  cent. 
Full-load  power-factor  =  0.91. 

The  required  estimation  of  the  full-load  current  may  be 
carried  out  as  follows: 

Horse-power  output  at  rated  load  =  200 

Watts  output  at  rated  load  =  200  X  746  =  149  200 

Efficiency  at  rated  load  =0.91 

1 4Q  200 

Watts  input  at  rated  load  -  =  164  000 

u.yi 


116       POLYPHASE  GENERATORS  AND  MOTORS 


Watts  input  per  phase  at  rated  load 
Power-factor  at  rated  load 


164  OOP 

3 
0.91 


=  54700 


54  700 

Volt-amp,  input  per  phase  at  rated  load  =  -777^-  =  60  200 

u.yi 

Pressure  between  terminals  (in  volts)        =  1000 
Phase  pressure 


Current  per  phase  at  rated  load 


60200 

577 


=  104 


TABLE  17. — PRELIMINARY  VALUES  OF  FULL  LOAD  EFFICIENCY,  IN  PER 
CENT,  FOR  POLYPHASE  SQUIRREL-CAGF  INDUCTION  MOTORS.  THE 
VALUES  GIVEN  CORRESPOND  TO  THOSE  OF  NORMAL  MOTORS. 


Rated 
Output 
in  Horse- 
power. 

Efficiencies  for  the  Following  Periodicities  and  Synchronous  Speeds. 

Periodicity  =12.5  Cycles. 

Periodicity  =25  Cycles. 

Periodicity  =50  Cycles. 

375 

188 

94 

750 

375 

188 

1500 

750 

375 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

10 

76.7 

72.8 

65.0 

82.5 

79.1 

74.6 

84.1 

80.6 

75.7 

20 

79.0 

74.7 

69.4 

84.0 

80.6 

76.6 

85.5 

82.6 

78.1 

30 

80.7 

77.3 

72.3 

85.0 

82.0 

78.1 

87.3 

84.1 

80.4 

40 

81.4 

78.0 

73.5 

86.5 

83.3 

79.8 

88.8 

86.0 

82.1 

50 

83.0 

80.0 

75.6 

87.5 

84.5 

81.1 

90.0 

87.3 

84.0 

60 

84.0 

81.1 

77.1 

88.0 

85.5 

82.5 

90.8 

88.3 

85.3 

80 

86.0" 

83.3 

80.1 

89.8 

87.3 

84.8 

92.5 

90.3 

.87.8 

100 

87.0 

85.0 

82.1 

91,3 

88.8 

86.5 

93.8 

91.8 

89.3 

300 

89.0 

87.0 

84.8 

92.5 

90.8 

88.8 

95.0 

93.5 

90.8 

500 

89.8 

88.0 

86.0 

93.2 

91.5 

89.8 

95.5 

94.0 

91.8 

700 

90.5 

88.8 

87.0 

93.8 

92.3 

90.5 

95.7 

94.3 

92.3 

1000 

91.0 

89.8 

88.5 

94.0 

92.8 

91.3 

95.8 

95.5 

92.8 

Conductors  per  phase.     For  a  preliminary  estimate  we  may 
proceed  as  follows: 


Conductors  per  phase 
Conductors  per  pole  per  phase 


24200 
104 

232 


=  232. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   111 


We  must  round  off  this  value  to  some  suitable  whole  number, 
say  39  or  40,  taking  whichever  leads  to  the  best  arrangement 
of  the  winding.  An  inspection  indicates  that  we  should  consider 
the  following  alternatives: 

40  conductors  arranged    8  per  slot  in  (~o~=  )  5  slots. 

"      «       ^2_\         ' 
U     / 

„       «       P     \o     „ 

(u  r 


40 
39 
40 


10 
13 
20 


TABLE  18. — PRELIMINARY  VALUES  FOR  FULL-LOAD  POWER  FACTOR  OF  POLY- 
PHASE SQUIRREL-CAGE  INDUCTION  MOTORS  OF  NORMAL  DESIGN. 


Power-factor  for  the  Following  Periodicities  and  Synchronous  Speeds. 

Rated 

Output 

Periodicity  =  12.5  Cycles. 

Periodicity  =25  Cycles. 

Periodicity  =50  Cycles. 

in  Horse- 

power. 

375 

188 

94 

750 

375 

188 

1500 

750 

375 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m. 

r.p.m 

10 

0.896 

0.820 

0.770 

0.882 

0.820 

0.760 

0.898 

0.830 

0.750 

20 

0.906 

0.829 

0.780 

0.892 

0.830 

0.770 

0.904 

0.840 

0.760 

30 

0.916 

0.838 

0.790 

0.900 

0.837 

0.780 

0.908- 

0.847 

0.770 

40 

0.921 

0.847 

0.800 

0.906 

0.844 

0.790 

0.914 

0.852 

0.780 

50 

0.928 

0.855 

0.810 

0.911 

0.850 

0.800 

0.917 

0.857 

0.790 

60 

0.931 

0.864 

0.820 

0.916 

0.856 

0.810 

0.921 

0.862 

0.800 

80 

0.941 

0.872 

0.830 

0.924 

0.862 

0.820 

0.926 

0.867 

0.810 

100 

0.947 

0.880 

0.840 

0.930 

0.871 

0.830 

0.928 

0.872 

0.820 

300 

0.955 

0.887 

0.850 

0.938 

0.880 

0.840 

0.931 

0.876 

0.830 

500 

0.960 

0.896 

0.860 

0.942 

0.886 

0.850 

0.932 

0.880 

0.840 

700 

0.962 

0.904 

0.870 

0.945 

0.892 

0.860 

0.932 

0.884 

0.850 

1000 

0.963 

0.912 

0.880 

0.946 

0.900 

0.870 

0.932 

0.888 

0.860 

Number  of  Slots  per  Pole  per  Phase.  There  cannoj;  be  given 
any  absolute  rule  as  regards  the  number  of  slots  per  pole  per 
phase  which  should  be  employed.  In  a  general  way  it  may  be 


118       POLYPHASE  GENERATORS  AND  MOTORS 

stated  that  the  quality  of  the  performance  of  the  motor  is  higher, 
the  greater  the  number  of  slots  per  pole  per  phase.  But  the 
overall  dimensions  and  the  weight  and  the  Total  Works  Cost 
increase  with  increasing  subdivision  of  the  winding  amongst 
many  slots,  and  consequently  the  designer  should  endeavor  to 
arrive  at  a  reasonable  compromise  between  quality  and  cost. 

The  Slot  Pitch.  We  may  designate  as  the  slot  pitch  the  dis- 
tance (measured  at  the  air-gap)  from  the  center  line  of  one  slot 
to  the  center  line  of  the  next  adjacent  slot.  Since  this  quantity 
is  usually  small,  it  is  generally  convenient  to  express  it  in  mm. 
Good  representative  values  for  the  stator  slot  pitch  are  given 
in  Table  19.  The  values  in  the  table  may  be  taken  as  applying 
to  designs  for  moderate  pressures.  The  higher  the  pressure, 
the  more  must  one  depart  from  the  tabulated  values  in  the  direc- 
tion of  employing  fewer  slots, 

TABLE  19. — VALUES  OF  STATOR  SLOT  PITCH  FOR  INDUCTION  MOTORS. 


T,  the  Polar  Pitch 
(in  cm.). 

Stator  Slot  Pitch,  (in  mm.) 

12 

15.0 

14 

16.4 

16 

17.7 

18 

18.9 

20 

20.0 

25 

22.3 

30 

24.4 

35 

26.0 

40 

27.0 

From  Table  19  we  select  as  appropriate  for  our  motor,  a 
trial  slot  pitch  of  about  25  mm.  Since  T  is  equal  to  325  mm., 
we  should  have: 


325 
25 


=  13  slots  per  pole. 


But  the  number  finally  chosen  should  be  a  multiple  of  3,  the 
number  of  phases.     Consequently  let  us  employ  12  slots  per  pole, 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  119 

(12     \ 
—  =  )  4  slots  per  pole  per  phase  (pppp).    Thus   we  must 
o     / 

revise  the  stator  slot  pitch  to: 


=  27.1  mm. 
\& 

The  Total  Number  of  Stator  Slots.  Since  our  machine  has 
6  poles,  we  shall  have: 

12X6  =  72  stator  slots. 

The  appropriate  slot  layout  will  be  based  on  10  conductors 
per  slot  and  : 

4X10  =  40  conductors  pppp. 
and 

6X40     10_, 

—  ^—  =  120  turns  in  series  per  phase. 
& 

Let  us  denote  the  turns  in  series  per  phase  by  T.     Then: 

^=120. 

THE  PRESSURE  FORMULA 

In  our  discussion  of  the  design  of  generators  of  alternating 
electricity  we  have  become  acquainted  with  the  pressure  formula  : 

V=KXTX~XM. 

In  this  formula  we  have: 

y  =  the  phase  pressure  in  volts; 
il  =  a  coefficient; 
T  =  turns  in  series  per  phase; 
~  =  periodicity  in  cycles  per  second; 
M  =  flux  per  pole  in  megalines. 

For  a  motor,  the  phase  pressure  in  the  above  formula  must, 
for  full  load,  be  taken  smaller  than  the  terminal  pressure,  to  the 
extent  of  the  IR  drop  in  the  stator  windings.  But  at  no  load 


120       POLYPHASE  GENERATORS  AND  MOTORS 

the  phase  pressure  is  equal  to  the  terminal  pressure  divided  by  V3. 
Therefore, 

1000 
Phase  pressure  =  :r-,™  =  577  volts. 


The  coefficient  K  depends  upon  the  spread  of  the  winding 
and  the  manner  of  distribution  of  the  flux.  For  the  conditions 
pertaining  to  a  three-phase  induction  motor  with  a  full-pitch 
winding  we  have: 

#  =  0.042. 

For  other  winding  pitches,  the  appropriate  value  of  K  may 
be  derived  by  following  the  rule  previously  set  forth  on  pp.  16 
to  18  of  Chapter  II,  where  the  voltage  formula  for  generators 
of  alternating  electricity  is  discussed. 

Since  our  motor  is  for  operation  on  a  25-cycle  circuit,  we  have: 


Thus  at  no  load  we  have: 

577  =  0.042X120X25XM; 
M  =  4.57  megalines. 

THE  MAGNETIC  CIRCUIT  OF  THE  INDUCTION  MOTOR 

In  Fig.  62  are  indicated  the  paths  followed  by  the  magnetic 
lines  in  induction  motors  with  2,  4,  and  8  poles.  One  object  of 
the  three  diagrams  has  been  to  draw  attention  to  the  dependence 
of  the  length  of  the  iron  part  of  the  path,  on  the  number  of 
poles.  Thus  while  in  the  2-pole  machine,  some  of  the  lines 
extend  over  nearly  a  semi-circumference,  in  the  stator  core  and 
in  the  rotor  core;  their  extent  is  very  small  in  the  8-pole  design. 
As  a  consequence,  the  sum  of  the  magnetic  reluctances  of  the 
air-gap  and  teeth  constitutes  a  greater  percentage  of  the  total 
magnetic  reluctance,  the  greater  the  number  of  poles. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   121 

Fig.  63  has  been  drawn  to  distinguish  that  portion  of  the  total 
flux  which  corresponds  to  one  pole.  It  is  drawn  to  correspond 
to  a  6-pole  machine. 


2  Pole 


4  Pole 


8  Pole 


FIG.  62. — Diagrammatic  Sketches  of  2-,  4-,  and  8-pole  Induction-motor  Cores, 
showing  the  Difference  in  the  Mean  Length  of  the  Magnetic  Path. 

At  this  stage  of  our  calculations,  we  wish  to  ascertain  the  cross- 
section  which  must  be  allowed  for  the  stator  teeth.  A  crest 
density  of  15  500  lines  per  sq.  cm.  is  appropriate  for  the  stator 


FIG.  63. — Diagrammatic  Representation  of  that  Portion  of  the  Total  Path 
which  Corresponds  to  One  Pole  of  an  Induction  Motor.  [The  heavy 
dotted  lines  indicate  the  mean  length  of  the  magnetic  path  for  one  pole.] 

teeth  in  such  a  design  as  that  which  we  are  considering.  A  skilled 
designer  will,  on  occasions,  resort  to  tooth  densities  as  high  as 
19  000,  but  it  requires  experience  to  distinguish  appropriate  cases 


122       POLYPHASE  GENERATORS  AND  MOTORS 

for  such  high  densities  and  the  student  will  be  well  advised  to 
employ  lower  densities  until  by  dint  of  practice  in  designing,  he  is 
competent  to  exercise  judgment  in  the  matter.  In  general  the 
designer  will  employ  a  lower  tooth  density  the  greater  the  number 
of  poles.  This  is  for  two  reasons:  firstly,  as  already  mentioned 
in  connection  with  Fig.  62,  the  magnetic  reluctance  of  the  air- 
gap  and  teeth  constitutes  a  greater  percentage  of  the  total  reluc- 
tance the  greater  the  number  of  poles;  and  secondly,  (for  reasons 
which  will  be  better  understood  at  a  later  stage),  a  high  tooth 
density  acts  to  impair  the  power-factor  of  a  machine  with  many 
poles,  to  a  greater  extent  than  in  the  case  of  a  machine  with  few 
poles. 

In  Figs.  64  and  65  are  shown  two  diagrams.  These  represent 
the  distribution  of  the  flux  around  the  periphery  of  our  6-pole 
motor  at  two  instants  one-twelfth  of  a  cycle  apart.  Since  the 
periodicity  is  25  cycles  per  second,  one-twelfth  of  a  cycle  occupies 

(          ^=  Wfoth  °f  a  second.     After  another  g^th  of  a  second 

the  flux  again  assumes  the  shape  indicated  in  Fig.  64,  but  dis- 
placed further  along  the  circumference,  as  indicated  in  Fig.  66. 
In  other  words,  as  the  flux  travels  around  the  stator  core,  its 
distribution  is  continually  altering  in  shape  from  the  typical 
form  shown  in  Fig.  64,  to  that  shown  in  Fig.  65,  and  back  to  that 
shown  in  Fig.  66  (which  is  identical  with  Fig.  64,  except  that  it 
has  advanced  further  in  its  travel  around  the  stator) .  Successive 
positions  of  the  flux,  each  ^th  second  later  than  its  predecessor, 
are  drawn  in  Figs.  67  to  70.  Comparing  Fig.  70  with  Fig.  64, 
we  see  that  they  are  identical  except  that  in  Fig.  64,  a  south 
flux  occupies  those  portions  of  the  stator,  which,  in  Fig.  64, 
were  occupied  by  a  north  flux.  In  other  words,  a  half  cycle  has 
occurred  in  the  course  of  the  (dhr^iroth  of  a  second  which  has 
elapsed  while  the  flux  has  traveled  from  the  position  shown  in 
Fig.  64,  to  that  shown  in  Fig.  70.  A  whole  cycle  will  have  occurred 
in  u^th  of  a  second  (the  periodicity  is  25  cycles  per  second) ;  and 
the  flux  will  then  have  been  displaced  to  the  extent  of  the  space 
occupied  by  one  pair  of  poles.  At  the  end  of  the  time  occupied 

by  3  cycles  (*\ths  of  a  second)  the  flux  will  have  completed  one 

/ (\     \ 
revolution  around  the  stator  core,  since  the  machine  has  ( 9  =  ) 

3  pairs  of  poles. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  123 


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g 

o 

S 

1.8 

1.6 
1.4 
1.2 
1.0 

0.8 
O.G 
0.4 
0.2 

^ 

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i 

\ 

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* 

0                 0.5                  1                  1.5                  2                  2.5                  3 
Cycles  per  Sec. 

i            I            1            i            i             i            i 

U                       50                       25                        50                       25                        10                       25 
Time  in  Seconds 

FlGS.  64  to  70.— Diagrams 
Flux  as  it 


Indicating  the  Variations  in  the  Shafce  of  the 
Around  the  Stator  Core. 


124       POLYPHASE  GENERATORS  AND  MOTORS 

In  estimating  the  magnetomotive  force  (mmf.)  which  must 
be  provided  for  overcoming  the  reluctance  of  the  magnetic  circuit, 
we  must  base  our  calculations  on  the  crest  flux  density.  This 
corresponds  to  the  flux  distributions  represented  in  Figs.  65, 
67  and  69.  It  can  be  shown  *  that  the  crest  density  indicated 
in  these  figures  is  1.7  times  the  average  density.  In  other 


FIG.  71. — Diagram  Illustrating  that  the  Crest  Density  in  the  Air-gap  and 
Teeth  of  an  Induction  Motor  is  1.7  Times  the  Average  Density. 


words,  the  crest  density  with  the  flux  distribution  correspond- 
ing to  the   peaked    curve  in  Fig.  71,  is  1.7  times  the    average 

*  This  is  demonstrated,  step  by  step,  on  pp.  380  to  390  of  the  2d  edition  of 
the  author's  "Electric  Motors"  (Whittaker  &  Co.,  London  and  New  York, 
1910;, 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  125 

density  indicated  by  the  rectangle  in  the  same  figure.  Each 
tooth  is,  in  turn,  located  at  the  center  of  the  rotating  flux  and  is 
in  turn  subjected  to  this  crest  density. 

Thus  for  our  assumption  of  a  crest  density  of  15  500  lines  per 
square  centimeter  we  shall  have : 

15  500 
Average  density  in  stator  teeth  =  — — ^—  =  9100  lines  per  sq.cm. 


Since  we  have  a  flux  of  4.57  megalines,  we  must  provide,  per 
pole,  a  tooth  cross-section  of: 


4570000     _nn 

=  500  sq.cm. 


There  are  12  stator  teeth  per  pole.     The  cross-section  of  each 
tooth  must  thus  be: 

—-  =  41.6  sq.cm. 


Before  we  can  attain  our  present  object  of  determining  upon  the 
width  of  the  tooth,  we  shall  have  to  digress  and  take  up  the 
matter  of  the  proportioning  of  the  ventilating  ducts. 


VENTILATING  DUCTS 

The  employment  of  a  large  number  of  ventilating  ducts  in 
the  cores  of  induction  motors,  renders  permissible,  from  the 
temperature  standpoint,  the  adoption  of  much  higher  flux  den- 
sities and  current  densities  than  could  otherwise  be  employed, 
and  thus  leads  to  a  light  and  economical  design. 

In  Table  20  are  given  rough  values  for  the  number  of  ducts, 
each  15  mm.  wide,  which  may  be  taken  as  suitable,  under  various 
circumstances  of  peripheral  speed  and  values  of  \g. 

In  our  case,  where  the  peripheral  speed  is  16.2  meters  per  second 
and  Xg  is  43,  the  table  indicates  1.8  ducts  per  dm.,  or  a  total  of 


126       POLYPHASE  GENERATORS  AND  MOTORS 


1.8X4.3  =  7.7    ducts,  to  be  a  suitable  value.     Eight   ducts  will 
be  employed  and  they  will  require  .8X1.5  =  12.0  cm. 

TABLE  20. — VENTILATING  DUCTS  FOR  INDUCTION  MOTORS. 


Peripheral  Speed  in 
Meters  per  Second. 

Number  of  Ventilating  Ducts  (Each  15  mm.  Wide),  which  Can 
Appropriately  be  Used,  per  Decimeter  of  \g. 

\g=W. 

X0=30. 

X0=50. 

10 

2.2 

2.3 

2.4 

15 

1.7 

1.9 

2.1 

20 

1.5 

1.7 

1.9 

25 

1.3 

1.5 

1.7 

30 

1.1 

1.3 

1.5 

35 

1.0 

1.2 

1.3 

40 

0.9 

1.1 

1.2 

The  varnish  by  means  of  which  adjacent  core  plates  are 
insulated  from  one  another,  will  occupy  some  10  per  cent  of  the 
total   depth  occupied   by  the  insulated 
core  plates. 

Thus  for  Xw,  the  net  core  length,  we 
arrive  at  the  value: 

Xn=(43-12)X0.9  =  3lX0.9  =  27.9  cm. 

WIDTH  OF  STATOR  TOOTH 

We  may  now  complete  the  calculation 
of  the  width  of  the  stator  tooth  at  its 
narrowest  part.  The  tooth  will  be  of 
the  form  indicated  in  Fig.  72,  and  the 
narrowest  part  (or  neck)  will  be  at  a 
diameter  not  appreciably  greater  than 

the  air-gap  diameter.     It  is  not  import- 
FIG.    72.  —  Stator  Tooth  r^  .   n  ,    .       ,, 

of  200  H.P.  Induction     ant  tO  be  sPecially  exact  m   the   matter> 

Motor,  so   let   us  make  the  rough  preliminary 

assumption  that  the    diameter    at    this 

narrowest  part  is  1  cm.  greater  than  D,  the  air-gap  diameter. 
We  have: 

D  =  62.0  cm. 


I      <-14.9mro-> 
h 22m 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   127 
Diameter  to  neck  of  tooth  =62.0+1.0  =  63.0  cm. 

Tooth  pitch  at  neck  =~72~  =^7.5  mm. 

Required  cross-section  of  tooth  at  neck  =  4 1.6  sq.cm. 

Width  of  tooth  at  neck  =  -= =  -7^77-  =  14.9  mm. 

An        279 

Width  of  Slot.     The  slot  will  have  parallel  sides,  and  its  width 
will  be: 

27.5 -14.9  =  12.6  mm. 


This  will  be  its  width  when  punched.  Owing  to  inevitable  slight 
inaccuracies  in  building  up  the  stator  core  from  the  individual 
punchings,  the  assembled  width  of  the  slot  will  be  some  0.3  mm. 
less,  or 

12.6-0.3  =  12.3  mm. 

This  allowance  of  0.3  mm.  is  termed  the  "  slot  tolerance." 

Having  now  determined  the  width  of  the  stator  slot,  it  would 
appear  in  order  to  proceed  at  once  to  determine  its  depth.  But 
this  depends  upon  the  copper  contents  for  which  space  must 
be  provided.  Consequently  we  must  now  turn  our  attention 
to  the  determination  of: 

The  Dimensions  of  the  Stator  Conductor.  The  current 
density  in  the  stator  conductor  is  determined  upon  as  a  compro- 
mise amongst  a  number  of  considerations,  one  of  the  chief  of 
which  is  the  permissible  value  of  the  watts  per  square  decimeter 
(sq.dm.)  of  peripheral  radiating  surface  at  the  air-gap.  This 
value  is  itself  influenced  by  such  factors  as  the  peripheral  speed 
and  the  ventilating  facilities  provided,  hence  the  current  density 
will  also  be  influenced  by  these  considerations. 

The  value  of  the  watts  per  square  decimeter  of  peripheral 
radiating  surface  at  the  air-gap,  cannot,  unfortunately,  be  ascer- 
tained until  a  later  stage  when  we  shall  have  determined  not 
only  the  copper  losses,  but  also  the  core  loss.  If,  at  that  later 
stage,  the  value  obtained  for  the  watts  per  square  decimeter 
of  peripheral  radiating  surface  at  the  air-gap,  shall  be  found  to 
be  unsuitable,  it  will  be  necessary  to  readjust  the  design. 


128       POLYPHASE  GENERATORS  AND  MOTORS 


Table  21  has  been  compiled  to  give  preliminary  representa- 
tive values  for  the  stator  current  density  for  various  outputs 
and  peripheral  speeds,  for  designs  of  normal  proportions. 

TABLE  21. — PRELIMINARY  VALUES  FOR  THE  STATOR  CURRENT  DENSITY. 


Rated  Output 
in  h.p. 

Current  Density  for  Various  Peripheral  Speeds  in  Meters  per 
Second  (mps). 

10  mps. 

20  mps. 

30  mps. 

40  mps. 

5 

400 

10 

380 

400 

50 

350 

370 

390 

100 

320 

340 

360 

380 

500 

290 

320 

340 

350 

1000 

280 

300 

310 

The  Slot  Insulation.  The  fact  that  the  slot  has  an  assembled 
width  of  12.3  mm.  is  not  to  be  taken  as  indicating  that  this  width 
is  available  for  the  conductors.  The  iron  core  must  be  separated 
from  the  conductors  by  a  considerable  thickness  of  insulation. 
This  insulation  consists  preferably  in  specially-manufactured 
tubes  of  high-grade  insulating  material.  Suitable  thicknesses 
are  indicated  in  Table  22. 

TABLE  22. — VALUES  FOR  THE  THICKNESS  OF  THE  STATOR  SLOT  LINING. 


Normal  Pressure  (in 
Volts)  for  Which  the 
Induction  Motor  is 
Wound. 

Thickness  of  Slot  Lining 
in  mm. 

500 

0.9 

1000 

1.4 

2000 

2.3 

3000 

2.9 

4000 

3.3 

6000 

4.0 

8000 

4.7 

10000 

5.2 

12000 

5.6 

In  Fig.  73  a  sketch  is  given  of  the  slot  with  the  insulating 
tube  in  place,  and  before  winding.     The  precise  depth  has  not 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   129 


yet  been  determined,  but  the  width  of  the  available  space  is  seen 
to  be  12.3  —  2X1.4  =  9.5  mm.  Thus  we  see  that  a  space  9.5  mm. 
wide  is  available  for  the  insulated  conductors. 

From  Table  21  we  ascertain  that  the  stator  conductor  should 
be  proportioned  for  about  300  amperes 
per  square  centimeter.     Since  we  have,  I— 12.3  mm— > 

for  the  full-load  current,  7=  104  amperes, 
the  cross-section  should  be : 


104 

——  =  0.347  sq.cm.  or  34.7  sq.mm. 

oUU 

In  the  interests  of  securing  a  suit- 
able amount  of  flexibility  in  the  process 
of  winding,  let  us  divide  this  aggregate 
cross-section  into  two  conductors  which 
shall  be  in  parallel  and  each  of  which 
shall  have  a  cross-section  of  some: 


—-  =  17.4  sq.mm. 


The  diameter  of  a  wire  with  a  cross- 
section  of  17.4  sq.mm.  is: 


D  = 


4.70  mm. 


«-9.5 


«-1.4mm 


FIG.  73.— Stator  Slot  of  200 
H.P.  Induction  Motor, 
showing  Insulating  Tube 
in  Place. 


The  bare  diameter  of  each  wire  would,  on  this  basis,  be 
4.70  mm. 

In  Table  23  are  given  the  thicknesses  of  insulation  on  suitable 
grades  of  cotton-covered  wires  employed  in  work  of  this  nature. 

We  see  that  our  wire  of  4.70  mm.  diameter  would,  if  double 
cotton  covered,  have  a  thickness  of  insulation  of  about  0.18mm. 
Consequently  its  insulated  diameter  would  be: 

4.70+2X0.18  =  4.70+0.36  =  5.06  mm. 

But  the  width  of  the  winding  space  is  seen  from  Fig.  73  to  be 
only  9.5  mm.     The  natural  arrangement  in  this  case,  would  be 


130       POLYPHASE  GENERATORS  AND  MOTORS 


to    place    the  two  components  side  by  side.     Consequently  the 
insulated  diameter  must  not  exceed: . 

-^-  =  4.75  mm. 
TASLE  23. — VALUES  OF  THE  THICKNESS  OF  COTTON  COVERING. 


Diameter  of  Bare 
Conductor  (in  mm.). 

Thickness  of  Insulation  (in  mm.). 

Single  Cotton 
Covered. 

Double  Cotton 
Covered. 

Triple  Cotton 
Covered. 

1 

0.060 

.       0.100 

2 

0.080 

0.127 

0.180 

3 

0.098 

0.150 

0.207 

4 

0.112 

0.167 

0.227 

5 

0.123 

0.183 

0.244 

6 

0.133 

0.196 

0.258 

8 

0.147 

0.214 

0.279 

10 

0.220 

0.288 

12 

0.290 

So  let  us  reduce  the  bare  diameter  to : 

4.75-0.36  =  4.39  mm. 
The  readjusted  conductor  has  a  cross-section  of  only: 

7X4.392  =  15.1  sq.mm. 

A 

The  two  component  conductors  make  up  a  cross-section  of: 
2X0.151  =  0.302  sq.cm. 

The  revised  current  density  is: 

104 
^-57^  =  345  amperes  per  sq.cm. 


This  density  is  considerably  higher  than  the  value  in  Table  21, 
but  we  have  made  a  rather  liberal  provision  for  ventilating  ducts 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   131 


and  there  is  no  reason,  at  this  stage,  to  anticipate  that  we  have 
exceeded  permissible  values.  However,  this  must  be  put  to  the 
test  at  a  later  stage  when  we  shall  have  sufficient  data  to  estimate 
the  temperature  rise  on  the  basis  of  the  watts  lost  per  square 
decimeter  of  peripheral  radiating  surface  at  the  air-gap. 

Since  we  must  provide  for   10  conductors   (20   component 
wires)  per  slot,  the  height 
of  the  winding  space  must 
be  at  least 

10X4.75  =  47.5  mm. 


But  it  will  be  impracti- 
cable to  thread  the  wires 
into  place  with  complete 
avoidance  of  any  lost  space. 
So  let  us  add  5  per  cent  to 
the  height  of  the  winding 
space,  making  it: 


L4mm 


10  Ins.  Conds.    47.5  mm 


1.4mm 


2.2mm 


2.5mm 
54.0mm 


1.05X47.5  =  50  mm. 

The  slot  with  the  wires 
in  place,  is  drawn  in  Fig. 
74.  It  is  seen  from  this 
figure  that  the  total  depth 
is  54  mm.  The  slot  open- 
ing is  6  mm.  wide. 

The  Slot  Space  Factor. 
The  total  area  of  cross- 
section  of  copper  in  the 
slot  amounts  to  10X0.302  = 
3.02  sq.cm. 

The  product  of  depth  and  punched  width  of  slot  is  equal  to 
1.26X5.40  =  6.80  sq.cm. 

3  02 

Space  factor  of  stator  slot  =  r1  ^  =  0.445. 

b.oU 

It  is  to  be  distinctly  noted  that  this  slot  design  is  merely  a 
preliminary  layout.  Should  it  at  a  later  stage  not  be  found  to 
fulfil  the  requirements  as  regards  sufficiently-low  temperature- 


FIG.  74.— Stator  Slot  of  200  H.P.  Motor 
with  Winding  in  Place. 


132       POLYPHASE  GENERATORS  AND  MOTORS 


rise  at  rated  load,  it  will  be  necessary  to  consider  ways  and  means 

of  so  modifying  the  design  as  to  fulfil  the  requirements. 

Preliminary  Proportions  for  the   Rotor   Slot.     For   reasons 

which  will  appear  later,  there  will  be  a  number  of  rotor  slots 
not  differing  greatly  from  the  number  of 
stator  slots  and  these  rotor  slots  will  be  of 
about  the  same  order  of  depth  as  the  stator 
slots,  but  considerably  narrower.  The  result 
will  be  that  the  rotor  tooth  density  will  be 
fully  as  low  or. even  lower  than  the  stator 
tooth  density.  Let  us  for  the  present,  con- 
sider that  the  rotor  slots  are  54  mm.  deep 
and  that  the  crest  density  in  the  rotor  teeth 
is,  at  no  load,  15  500  lines  per  square  cen- 
timeter. 

Let  us  further  assume  for  the  present 
that  the  rotor  slots  are  nearly  wide  open, 
the  shape  of  a  rotor  slot  being  somewhat  as 

FIG.  75. — Rotor  Slot  indicated   in   Fig.    75.      In   the   final  design, 

of  200  H.P.Squirrel-  the    width   of   the   rotor    slot    opening   may 

cage  Motor.  readily  be  so  adjusted  as  to  constitute  about 

20  per  cent  of  the   rotor  tooth  pitch   at  the 

surface  of  the  rotor,  the  tooth  surface  thus    constituting  some 

80  per  cent  of  the  tooth  pitch. 

Determination    of    Cross-section    of   Air-gap.      The    stator 

tooth  pitch  at  the  air-gap  is: 


620  Xic 

72 


=  27.1  mm. 


Stator  slot  opening  =  6.0  mm. 
The  stator  tooth  surface  thus  constitutes: 

27  l  —  6  0 

'271     X 100  =  78.0  per  cent 

of  the  stator  slot  pitch. 

Considering  the  average  value  of  this  percentage  on  both 
sides  of  the  air-gap,  we  find  it  to  be : 

78.0+80.0 

—  =  79.0  per  cent. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   133 


If  there  were  no  slot  openings,  the  cross-sections  of  the  surfaces 
from  which  the  lines  of  each  pole  emerge  and  into  which  they 
enter,  could  be  taken  as  Xn  X  T. 

But  the  slot  openings  bring  this  cross-section  down  to 

0.79XXnX-r. 

In  crossing  the  air-gap,  however,  the  lines  spread  out  and 
may  be  considered  to  occupy  a  greater  cross-section  when  half 
way  across.  They  then  gradually  converge  as  they  approach 
the  surfaces  of  the  teeth  at  the  other  side  of  the  gap.  To  allow 
for  this  spreading,  we  may  increase  the  cross-section  by  15  per 
cent,  bringing  it  up  to  1.15X0.79XXnXT. 

For  our  200-h.p.  induction  motor  we  have: 

\n  =  27.9  cm.,     T  =  32.5  cm. 

Cross-section  of  air-gap  =  1.15X0.79X27.9X32.5  =  825  sq.cm. 

«     4  570  000 

Average  air-gap  density  (at  no  load)  =  — 5^ —  =  5550  lines   per 


sq.  cm. 


825 


Crest  density  =  1 .7  X  5550  =  9450. 


Radial  Depth  of  Air-gap.      Let  us  denote  the  radial  depth 
of  the  air-gap  in  mm.  by  A. 

Appropriate  preliminary  values  for  A  are  given  in  Table  24. 

TABLE  24. — APPROPRIATE  VALUES  FOR  A  THE  RADIAL  DEPTH  OF  THE  AIR- 
GAP  FOR  INDUCTION  MOTORS. 


D,  the  Air-gap 
Diameter, 

A,  the  Radial  Depth  of  the  Air-gap  (in  mm.),  for  Various  Values  of 
the  Peripheral  Speed  in  Meters  per  Second      (mps.). 

(in  cm.). 

10  mps. 

20  mps. 

30  mps. 

40  mps. 

20 

0.65 

0.75 

0.87 

1.00 

40 

0.87 

1.05 

1.25 

1.45 

60 

1.10 

1.35 

1.70 

1.90 

80 

1.30 

1.7 

2.0 

2.3 

100 

1.57 

2.0 

2.8 

2.8 

120 

1.77 

2.3 

2.8 

3.3 

134       POLYPHASE  GENERATORS  AND  MOTORS 

i 

For  our  200-h.p.  motor,  the  air-gap  diameter  is  62  cm.  and 
the  peripheral  speed  is  16.2  mps.  Consequently  the  radial  depth 
of  the  air-gap  is  A  =  1.3  mm. 

Preliminary  Magnetic  Data  for  Teeth  and  Air-gap.  We 
have  now  obtained  (or  assumed)  the  densities  in  the  teeth  and  in 
the  air-gap  and  we  have  the  lengths  of  these  portions  of  the  mag- 
netic circuit.  These  data  are: 


Length  (in  cm.)  of  Portions 
of  Magnet  Circuit. 

Crest  Density  at  no  Load 
(in  Lines  per  sq.cm.). 

Stator  teeth  
Rotor  teeth 

5.4 
5  4 

15500 
15500 

Air-gar) 

0.13 

9450 

The  magnetomotive  force  (mmf.)  required  to  overcome  the 
reluctance  of  the  above-tabulated  portions  of  the  magnetic 
circuit,  will,  in  most  designs,  constitute  a  predominatingly-large 
percentage  of  the  total  mmf.  As  the  designer  gains  experience, 
he  will  often  be  able,  after  calculating  this  portion  of  it,  safely 
to  use  his  judgment  in  assigning  (without  detailed  calculations), 
a  suitable  and  relatively-small  amount,  to  provide  for  the  mmf. 
required  for  overcoming  the  reluctance  of  the  stator  and  rotor 
cores.  In  the  great  majority  of  cases,  this  further  amount  con- 
stitutes so  small  a  percentage  of  the  total  mmf.  per  pole,  that 
a  very  considerable  divergence  from  the  value  which  would  be 
obtained  by  detailed  calculations,  would  not  seriously  influence 
the  total.  That  the  length  of  the  magnetic  circuit  in  the  stator 
and  rotor  cores  is  much  less  the  greater  the  number  of  poles  has 
already  been  pointed  out  on  page  120  and  has  been  illustrated 
by  the  sketches  of  the  2-,  4-,  and  8-pole  magnetic  circuits  in 
Fig.  62. 

Until,  however,  considerable  experience  has  been  gained  in 
calculating  the  magnetic  circuits  of  induction  motors,  it  is  not 
desirable  to  trust  to  obtaining  sufficient  accuracy  by  the  process 
of  multiplying  by  a  suitable  factor  the  mmf.  required  for  the  teeth 
and  the  air-gap.  It  is,  however,  not  necessary  to  consume  time 
in  accurately  estimating  the  mean  length  of  the  magnetic  path  in 
the  stator  and  rotor  cores.  On  the  contrary,  it  suffices  to  adopt  the 
assumption  that  the  portions  corresponding  to  one  pole  may  be 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  135 


taken  as  equal  to  the  circumferences  corresponding  to  the  mean 
core  diameter  of  the  stator  and  rotor  respectively,  divided  by  2P, 
where  P  is  the  number  of  poles.  But  as  a  step  toward  obtaining 
these  values,  it  is  necessary  to  obtain  the  external  diameter  of 
the  stator  core  discs  and  the  internal  diameter  of  the  rotor  core 
discs.  These  data,  in  turn,  are  dependent  upon  the  densities 
which  should  be  employed  behind  the  slots  in  the  stator  and  rotor 
respectively. 

Densities  in  Stator  and  Rotor  Cores.  The  densities  to  be 
employed  in  the  stator  and  rotor  cores  are  quantities  which  may 
be  varied  between  wide  limits.  In  general,  however,  the  core 
densities  should  be  lower,  the  greater  the  value  of  T  the  polar  pitch, 
and  ~  the  periodicity.  For  preliminary  assumptions,  the  values 
given  in  Table  25  will  be  found  suitable. 

TABLE  25. — DENSITIES  IN  STATOR  AND  ROTOR  CORES. 


Periodicity  in  Cycles 
per  Second. 

Values  of  Polar 
Pitch  (in  cm.)  . 

Density  in  Magnetic  Lines  per  sq.cm. 

Stator. 

Rotor. 

i,    j 

15 
20 
30  and  larger 

11500 
10  500 
10000 

14000 
13500 
13000 

,    { 

15 
20 

30  and  larger 

10000 
9000 
8500 

13000 
12500 
12000 

In  the  present  instance  we  find  from  the  above  table  that  for 
a  periodicity  of  25  cycles  per  second  and  a  polar  pitch  of  32.5 
cm.,  the  stator  density  should  be  10  000  lines  per  square  centimeter, 
and  the  rotor  density  13  000  lines  per  square  centimeter. 

Since  the  total  flux  (at  no  load)  is  4.57  megalines  per  pole, 
the  cross-sections  required  in  the  stator  and  rotor  cores  are 
respectively : 


Cross-section  of  stator  core  = 


4  570  OOP 
2X10000 


229  sq.cm. 


^  ..        ,  4570000     1I7. 

Cross-section  of  rotor  core  =         0  Artn  =  176  sq.cm. 

^  X  J-O  UUU 

~kn  is  equal  to  27.9  cm. 


136       POLYPHASE  GENERATORS  AND  MOTORS 

Consequently : 

Radial  depth  of  stator  punchings  (exclusive  of  slot  depth) 

229 


Radial  depth  of  rotor  punchings  (exclusive  of  slot  depth) 

176 
27.9 


=  6.3  cm. 


External  diameter  of  stator  punchings  =  62.0+2X5.4+2X8.2 

=  62.0+10.8+16.4 
=  89.2  cm. 

Internal  diameter  of  rotor  punchings 

=  62.0-2X0.13-2X5.4-2X6.3 
=  62.0-0.26-10.8-12.6 
=  38.3  cm. 

Diameter  at  bottom  of  stator  slots  =  62.0+2X5.4 

=  72.8  cm. 

Diameter  at  bottom  of  rotor  slots  =  62.0 -2X0.13 -2X5.4 

=  50.9  cm. 

89.2+72.8 
Mean  diameter  of  stator  core  — ~ =81.0  cm. 

Mean  diameter  of  rotor  core  '—^ —  L  =44.6  cm. 

Length  of  sta.  mag.  circ.  per  pole    =     '      -=21.2  cm. 

z  X  o 

Length  of  rotor  mag.  circ.  per  pole=      '-  =  11.7  cm. 

•^  X  0 

Compilation  of  Diameters.     It  is  of  interest  at  this  stage  to 
draw  up  an  orderly  list  of  the  leading  diameters: 

External  diameter  of  stator  core 892  mm. 

Diameter  at  bottom  of  stator  slots 728  mm. 

Internal  diameter  of  stator  (D) 620  mm. 

External  diameter  of  rotor  (D  —  2A) 617.4  mm. 

Diameter  at  bottom  of  rotor  slots 509  mm. 

Internal  diameter  of  rotor  core. .  .  383  mm. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   137 


Sketch  of  Magnetic  Portions  of  Design.  We  are  now  in  a 
position  to  make  a  preliminary  outline  drawing  of  the  magnetic 
parts  of  our  machine.  This  has  been  done  in  Fig.  76  which 
shows  the  leading  dimensions  of  the  magnetic  portions  of  the 
design  and  indicates  the  locations  of  the  end  connections  of  the 
stator  windings. 

Magnetic  Reluctance  of  Sheet  Steel.  In  the  design  of  the 
induction  motor,  our  magnetic  material  is  exclusively  sheet  steel. 
For  this  material  the  mmf.  data  in  the  first  two  columns  of  the 
table  previously  given  on  page  33  will  give  conservative  results. 

Tabulated  Data  of  Magnetic  Circuit.  We  now  have  the 
lengths  of  the  magnetic  paths,  the  densities,  and  also  data  for 
ascertaining  the  mmf.  required  at  all  parts  of  the  magnetic  circuit. 

Thus  for  example: 

Density  in  stator  core  =  10  000  lines  per  square  centimeter; 
Corresponding  mmf.  from  column  2  of  table  on  page  33 

=  4.6  ats.  per  centimeter 

Length  of  magnetic  circuit  in  stator  core  (per  pole)  =21.2  cm. 
Mmf.  required  for  stator  core  =  4.6X21. 2  =  98  ats. 

As  a  further  illustration  we  may  give  the  calculation  of  the 
mmf.  required  for  the  air-gap: 
Crest  density  in  air-gap  =  9450  lines  per  sq.cm. 

Corresponding  mmf.  =—X  9450  =  0.8X9450  =  7550  ats.  per  cm. 

Length  of  magnetic  circuit  in  air-gap  =0.13  cr  . 

Mmf.  required  for  air-gap  =  7550X0. 13         =980  ats. 

These  illustrations  will  suffice  to  render  clear  the  arrangement 
of  the  calculations  in  the  following  tabulated  form: 

TABLE  26. — ARRANGEMENT  OF  MMF.  CALCULATIONS. 


Part. 

U) 

Length  of  Mag. 
Circ.  in  cm. 

Density  in  Lines 
per  sq.cm. 

,<*) 

mmf.  per  cm. 

(AXB) 

Total  mmf. 

Stator  teeth.  . 
Rotor  teeth  .  . 
Air-gap 

5.4 
5.4 
0  13 

15500 
15500 
9450 

22 
22 
7550 

119 
119 
980 

Stator  core  .  . 
Rotor  core.  .  . 

21.2 
11.7 

10000 
13000 

4.6 
9.5 

98 
111 

Total  mmf 

.  per  Dole 

1427  ats. 

138       POLYPHASE  GENERATORS  AND  MOTORS 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  139 

(980  \ 

since  Tjoy  X 100  =  68.5  j,  the  air-gap  mmf. 

is  68.5  per  cent  of  the  total  required  mmf.     The  mmf.  required 
for  air-gap  and  teeth  is: 

980+119+119^.,™     1218     inA 

1427       -  X  100  =  ^^X100  =  85.5  per  cent 


of  the  total  required  mmf.  Attention  is  drawn  to  these  percent- 
ages to  bear  out  the  correctness  of  the  assertion  on  page  134 
that  considerable  inaccuracy  in  the  estimation  of  the  mean  length 
of  the  magnetic  circuit  in  the  stator  and  rotor  cores  will  not 
seriously  affect  the  accuracy  of  the  result  obtained  for  the  total 
mmf.  per  pole.  Consequently  the  use  of  the  rough  but  time- 
saving  rule  to  divide  by  twice  the  number  of  poles,  the  mean 
periphery  of  these  cores,  is  shown  to  be  justified. 

Resultant  mmf.  of  the  Three  Phases  Equals  Twice  the  mmf. 
of  One  Phase.  It  is  a  property  of  the  three-phase  windings  of 
induction  motors  that  the  resultant  mmf.  of  the  three  phases  is 
twice  that  exerted  by  one  phase  alone.  Consequently  in  our 

1427 
design,  each  phase  must  contribute  a  mmf.  of  —75—  =  714  ats. 

2i 

In  our  design,  T,  the  number  of  turns  in  series  per  phase,  is 

120 
equal  to  120.     The  design  has  6  poles.     Thus  we  have  -^-  =  20 

turns  per  pole  per  phase. 

Magnetizing  Current.  The  magnetizing  current  per  phase 
which  will  suffice  to  provide  the  required  714  ats.  must  obviously 
amount  to  : 

714 

—  =  35.7  crest  amperes 


or 


35  7 

—=  =  25.2  effective  amperes. 


Since  the  full-load  current  is  104  amperes,  the  magnetizing  cur- 

25  2 
rent  is  -—rX  100  =  24.  2  per  cent  of  the  full-load  current. 


140       POLYPHASE  GENERATORS  AND  MOTORS 

No-load  Current.  The  no-load  current  is  made  up  of  two 
components,  the  magnetizing  current  and  the  current  correspond- 
ing to  the  friction  and  windage  loss  and  the  core  loss,  i.e.,  to  the 
energy  current  at  no  load.  It  may  be  stated  in  advance  that  the 
energy  current  at  no  load  is  almost  always  very  small  in  compar- 
ison with  the  magnetizing  current.  Since,  furthermore,  the  mag- 
netizing current  and  the  energy  current  differ  from  one  another 
in  phase  by  90  degrees,  it  follows  that  their  resultant,  the  no- 
load  current,  will  not  differ  in  magnitude  appreciably  from  the 
magnetizing  current.  The  calculation  of  the  energy  current  is 
thus  a  matter  of  detail  which  can  well  be  deferred  to  another 
stage.  But  to  emphasize  the  relations  of  the  quantities  involved, 
let  us  assume  that  the  friction,  windage  and  core  loss  of  this  motor 
will  later  be  ascertained  to  be  a  matter  of  some  4500  watts.  This 

(1000     \ 
—  j=-  J577 

volts.     Consequently  the  energy  component  of  the  current  con- 

1500 
sumed  by  the  motor  at  no  load  is  -^==  =  2.6  amperes. 


The  no-load  current  thus  amounts  to  V25.22+2.62  =  25.3  amperes- 
In  other  words  the  no-load  current  and  the  magnetizing  current 
differ  from  one  another  in  magnitude  by  less  than  one-half  of 
one  per  cent,  in  this  instance.  Although  of  but  slight  practical 
importance,  it  may  be  interesting  to  show  that  they  differ  quite 
appreciably  in  phase.  For  we  have  for  the  angle  of  phase  differ- 
ence between  the  no-load  current  and  the  magnetizing  current: 

9  fin 
tan-1  ="    tan-1  0.103  =  5.9°. 


Thus  the  current  in  this  motor  when  it  is  running  unloaded, 
lags  (90  —  5.9  =  )  84.1°  behind  the  pressure;  in  other  words,  its 
power-factor  is  equal  to  (cos  84.1°  =  )  0.104. 

y,  the  Ratio  of  the  No-load  Current  to  the  Full-load  Current. 
It  is  convenient  to  designate  by  y,  the  ratio  of  the  no-load  current 
to  the  full-load  current.  For  our  motor  we  have: 

T-Sg-0.242. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR    141 

Practical  use  will  be  made  of  this  ratio  at  a  later  stage  in  the  design 
of  this  motor. 

H,  the  Average  Number  of  Slots  per  Pole.  It  is  also  con- 
venient to  adopt  a  symbol  for  the  average  of  the  number  of  slots 
per  pole  on  the  stator  and  rotor.  We  have  definitely  determined 
upon  the  use  of  12  slots  per  pole  on  the  stator.  We  have  reserved 
to  a  later  stage  of  the  calculations  the  determination  of  the  precise 
number  of  rotor  slots.  However  it  may  here  be  stated  that  it 
is  preferable  to  employ  a  number  of  rotor  slots  not  widely  differ- 
ing from  the  number  of  stator  slots.  Thus  as  a  preliminary 
assumption  we  may  take  12  as  the  average  of  the  numbers  of  slots 
per  pole  on  statoi  and  rotor.  Designating  this  quantity  by  H 
we  have ; 


THE    CIRCLE   RATIO 

We  have  now  all  the  necessary  data  for  determining  a 
quantity  for  which  we  shall  employ  the  symbol  a  and  which, 
for  reasons  which  we  shall  come  to  understand  as  we  proceed, 
we  shall  term  the  "  circle  ratio."  This  quantity  is  of  great 
utility  to  the  practical  designer.  Although  a  cannot  be  pre- 
determined with  any  approach  to  accuracy,  it  so  greatly 
assists  one's  mental  conceptions  from  the  qualitative  stand- 
point as  to  make  ample  amends  for  its  quantitative  uncertainty. 

We  have  seen  that  at  no  load,  the  current  consumed  by  an 
induction  motor  lags  nearly  90  degrees  behind  the  pressure. 
Let  us  picture  to  ourselves  a  motor  with  no  friction  or  core  loss 
and  with  windings  of  no  resistance.  In  such  a  motor  the  no- 
load  current  would  be  exclusively  magnetizing  and  would  lag  90 
degrees  behind  the  pressure. 

Let  us  assume  a  case  where,  at  no  load,  the  current  is  10 
amperes.  The  entire  magnetic  flux  emanating  from  the  stator  wind- 
ings will  cross  the  zone  occupied  by  the  secondary  conductors  (i.e., 
the  conductors  on  the  rotor)  and  pass  down  into  the  rotor  core. 
If  the  circumstance  of  the  presence  of  load  on  the  motor  were  not 
to  disturb  the  course  followed  by  the  magnetic  lines,  then  the 
magnetizing  component  of  the  current  flowing  into  the  motor 


142       POLYPHASE  GENERATORS  AND  MOTORS 

would  remain  the  same  with  load  as  it  is  at  no  load.  If  the  motor 
were  for  100  volts  per  phase,  then,  in  this  imaginary  case  where 
the  flux  remains  undisturbed  as  the  load  comes  on,  we  could  cal- 
culate in  a  very  simple  way  the  current  flowing  into  the  motor  for 
any  given  load.  To  illustrate;  let  us  assume  that  a  load  of  3000 
watts  is  carried  'by  this  hypothetical  motor.  A  load  of  3000  watts 
corresponds  to  an  output  of  1000  watts  per  phase.  Assuming  a 
motor  with  no  internal  losses,  the  input  will  also  amount  to  1000 
watts  per  phase.  Since  the  pressure  per  phase  is  100  volts, 
the  energy  component  of  the  current  input  per  phase  is 

(1000     \ 
-r^r  =  j  10  amperes.     Since  the  magnetizing  component  is  10 

amperes  the  resultant  current  per  phase  is  Vl02+102  =  14.1 
amperes.  The  vector  diagram  corresponding  to  these  conditions 
is  given  in  Fig.  77.  The  resultant  current  lags  behind  the 
terminal  pressure  by 


tan"1  jptan-1  1.0  =  45°. 

The  power-factor  is  : 

cos  45°  =  0.707. 

Let  us  double  the  load.     The  energy  component  of  the  cur 
rent  increases  to  2X10  =  20  amperes,  as  shown  in  Fig.  78. 
The  total  current  increases  to 


Vl024-202  =  22.4  amperes. 
The  angle  of  lag  becomes : 


tan-1  ijptan-1  0.5  =  26.6°. 


The  power-factor  increases  to: 

cos  26.6°  =  0.894. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  143 


Let  us  again  double  the  load,  thereby  increasing  the  energy 
component  of  the  current  to  40  amperes,  and  the  angle  of  lag 
to  14.0°  as  shown  in  Fig.  79.  The  total  current  is  now  41.3 
amperes  and  the  power-factor  is: 


740 .0^ 
Ul.3 


0.97. 


10 


FIG.  77 


FIG.  78 


FIG.  79 


FIGS.  77  to  79. — Vector  Diagrams  Relating  to  a  Hypothetical  Polyphase 
Induction  Motor  without  Magnetic  Leakage. 


At  this  rate  we  should  quickly  approach  unity  power-factor. 
The  curve  of  increase  of  power-factor  with  load,  would  be  that 
drawn  in  Fig.  80.  It  is  to  be  especially  noted  that  in  the  diagrams 
in  Figs.  77,  78  and  79,  the  vertical  ordinates  indicate  the  energy 


144       POLYPHASE  GENERATORS  AND  MOTORS 

components  of  the  total  current  and  the  horizontal  ordinates 
indicate  the  wattless  (or  magnetizing)  components  of  the  current. 


10      12       14       16       18      20      22      24 
Output  in  Kilowatts 


26 


FIG.  80.  —  Curve  of  Power-factor  of  Hypothetical  Polpyhase  Induction  Motor 
without  Magnetic  Leakage. 


It  would  be  very  nice  if  we  could  obtain  the  conditions  indicated 
in  the  diagrams  of  Figs.  77,  78  and  79.  That  is  to  say,  it  would 
be  very  nice  if  the  magnetizing  component  of  the  current  remained 


FIG.  81. — Diagrammatic  Representation  of  the  Distribution  of  the  Magneto- 
motive Forces  in  the  Stator  and  Rotor  Windings  of  a  Three-phase 
Induction  Motor. 


constant  with  increasing  load.     But  this  is  not  the  case.     As  the 
load  increases,  the  current  in  the  rotor  conductors  (which  was 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  145 

negligible  at  no  load),  increases.  The  combined  effect  of  the  cur- 
rent in  the  stator  and  rotor  conductors  is  to  divert  a  portion  of 
the  flux  out  of  the  path  which  it  followed  at  no  load.  In  Fig. 
81  are  indicated  diagramatically  a  few  slots  of  the  stator  and  rotor 
windings.  Considering  the  12  left-hand  conductors,  the  current  is 
indicated  as  flowing  (at  the  moment)  down  into  the  plane  of  the 
paper  in  the  stator  conductors  and  up  out  of  the  plane  of  the  paper 
in  the  rotor  conductors.  This  has  the  same  effect  (as  regards  the 
resultant  mmf.  of  the  stator  and  rotor  conductors),  as  would  be 
occasioned  by  the  arrangement  indicated  in  Fig.  82,  in  which 
the  stator  and  rotor  conductors  constitute  a  single  spiral. 
Obviously  the  mmf.  of  this  spiral  would  drive  the  flux  along  the 
air-gap  between  the  stator  and  rotor  surfaces.  The  reluctance 
of  the  circuit  traversed  by  the  magnetic  flux  thus  increases  grad- 
ually (with  increasing  current  input),  from  the  relatively  low 
reluctance  of  the  main  magnetic  circuit  traversed  by  the  entire  flux 
at  no  load,  up  to  the  far  higher  reluctance  of  the  circuit  traversed 
by  practically  the  entire  magnetic  flux  with  the  rotor  at  stand- 
still, the  pressure  at  the  terminals  of  the  stator  windings  being 
maintained  constant  throughout  this  entire  range  of  conditions. 
Consequently  the  magnetizing  component  of  the  current  consumed 
by  the  motor  increases  as  the  load  increases.  Thus  instead  of 
the  diagrams  in  Figs.  77,  78  and  79,  we  should  have  the  three 
diagrams  shown  at  the  right  hand  in  Fig.  83.  The  corresponding 
diagrams  at  the  left  hand  in  Fig.  83  are  simply  those  of  Figs. 
77,  78  and  79  introduced  into  Fig.  83  for  comparison.  In  both 
cases,  the  no-load  current  is  10  amperes.  But  in  the  practical  case 
with  magnetic  leakage,  the  loads  calling  respectively  for  energy 
components  of  10,  20  and  40  amperes  (loads  of  3000,  6000  and 


FIG.  82. — Diagram  Indicating  a  Solenoidal  Source  of  mmf.  Occasioning  a 
Flux  Along  the  Air-gap,  Equivalent  to  the  Leakage  Flux  in  an  Induction 
Motor. 


12000  watts)  involve  magnetizing  components  of  10.6,  12.1  and 
18.2  amperes. 


146       POLYPHASE  GENERATORS  AND  MOTORS 


10.0 


10.6 


10.0 


12,1 


10.0 


18.2 


No  Magnetic  Leakage 


Magnetic  Leakage 


FIG.  83. — Vector  Diagrams  for  Hypothetical  Motor  without  Magnetic  Leakage 
(at  Left),  and  for  Actual  Motor  with  Magnetic  Leakage  (at  Right). 

The  total  current  inputs  are  increased  as  follows : 


Load  (in  Watts). 

Current  Input  per  Phase. 

On  Assumption  of  no 
Magnetic  Leakage. 

On  Assumption  of  Magnetic 
Leakage. 

0 
3000 
6000 
12000 

10.0 
14.1 
22.4 
41.3 

10.0 
14.6 
23.4 
44.0 

POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  147 
The  power-factors  for  the  two  cases  are: 


Load  in  Watts. 

Power-factor. 

No  Magnetic  Leakage. 

Magnetic  Leakage. 

0 

0 

0 

3000 

0.707 

-  .(££-)  0.685 

6000 

0.894 

/20.0     \ 

\23.4~/°'855 

12000 

0.970 

£nr^-91° 

The  curve  of  power-factor  for  the  case  with  magnetic  leakage 
is  given  in  Fig.  84,  and  there  has  been  reproduced  dotted  in  this 


2        4 


10       12       14       16       18       20       22       24 
Output  in  Kilowatts 


FIG.  84. — Curves  of  Power-factor  of  Hypothetical  Induction  Motor  without 
Magnetic  Leakage  (Dotted  Line)  and  of  Actual  Motor  'with  Magnetic 
Leakage  (Full  Line). 

figure   the  power-factor   curve  for   the   case   with  no  magnetic 
leakage  which  has  already  been  given  in  Fig.  80. 


148       POLYPHASE  GENERATORS  AND  MOTORS 


In  Fig.  85,  the  hypothenuses  of  the  right-hand  diagrams 
of  Fig.  83  have  been  superposed,  and  their  right-hand  extremities 
are  seen  to  lie  upon  the  circumference  of  a  semi-circle  with  a 
diameter  of  200  amperes.  In  Fig.  85  the  magnetizing  current 
of  10  amperes  is  denoted  by  A  B.  The  diameter  of  the  semi- 
circle is  BD.  We  have: 


L  _ 


10    20    30    40     50    CO    70     80    90   100          120  130  140  150  160  170  180  190  200  210 

Wattless  Components  of  the  Current 

FIG.  85. — Circle  Diagram  for  a  Polyphase  Induction  Motor  with  a  No-load 
Current  of  10  Amperes  and  a  Circle-ratio  of  0.050. 

The  quantity  which  we  termed  the  "  circle  ratio  "  and  which 
we  designated  by  the  symbol  c,  is  the  ratio  of  AB  to  BD.  For 
this  case  we  have: 

AJJ-10-0050- 

B5-266-0-050' 

a  =  0.050. 

The  semi-circle  in  Fig.  85  is  the  locus  of  the  extremities  of  the 
vectors  representing  the  current  flowing  into  the  stator  winding. 
If,  for  any  value  of  the  current  input,  we  wish  to  ascertain  its 
phase  relations,  we  draw  an  arc  with  A  as  a  center  and  with  the 
value  of  the  current  as  a  radius.  The  intersection  of  this  arc 
with  the  semi-circle,  constitutes  one  extremity  of  the  vector 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   149 

representing  the  current,  and  A  constitutes  its  other  extremity. 
The  horizontal  projection  of  this  vector  is  its  wattless  component 
and  the  vertical  projection  is  its  energy  component.  Consequently 
we  also  have  the  convenient  relation  that  the  vertical  component 
is,  for  constant  pressure,  a  direct  measure  of  the  power  absorbed 
by  the  motor.  We  shall  give  further  attention  to  these  important 
relations  at  a  later  stage. 

The  circle  ratio  is  a  function  of  \g,  T,  A  and  H.  Knowing 
Xgr,  a,  A  and  H  we  can  obtain  a  rough  value  for  a  for  any  motor. 
In  other  words,  having  selected  these  four  quantities  for  a  motor 
which  we  are  designing,  we  can  obtain  a.  Having  calculated 
A  B,  the  magnetizing  current,  by  the  methods  already  set  forth 
on  page  139,  we  may  divide  it  by  a  and  thus  obtain  BD.  I  For 
we  have: 


We  are  now  in  a  position  to  construct,  for  the  200  h.p.  motor 
which  we  are  designing,  a  diagram  of  the  kind  represented  in 
Fig.  85. 

We  must  first  determine  a.     We  have: 


T  =  32.5  cm.; 
A  =  1.3  mm.; 
H  =  12. 

Knowing  these  four  quantities,  the  "  circle  factor,"  a,  may  be 
obtained  from  Table  27.  For  our  motor  we  find  from  the  table, 

a  =  0.041. 

The  values  of  a  in  Table  27  apply  to  designs  with  intermediate 
proportions  as  regards  slot  openings.  Should  both  stator  and 
rotor  slots  be  very  nearly  closed  (say  1  mm.  openings),  the  value 
of  a  would  be  increased  by — say — 20  per  cent  or  more.  On  the 
other  hand,  were  both  stator  and  rotor  slots  wide  open,  a  would 
be  decreased  by — say — some  20  per  cent  below  the  values  set  forth 
in  the  table.  It  cannot  be  too  strongly  emphasized  that  we  can- 
not predetermine  a  at  all  closely.  We  can,  however,  take  O.C41 
as  a  probable  value  for  a  in  the  case  of  our  design.  If,  on  test, 
the  observed  value  were  found  to  be  within  10  per  cent  of  0.041, 


150       POLYPHASE  GENERATORS  AND  MOTORS 


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POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   151 


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152       POLYPHASE  GENERATORS  AND  MOTORS 


the  result  should  be  considered  to  be  as  close  as  could  reasonably 
be  expected. 

This  indication  of  our  inability  to  closely  predetermine  a 
should  not  lead  to  a  disparagement  of  its  utility.  Most  practical 
purposes  are  amply  satisfied,  in  commercial  designing,  when  we 
can,  in  our  preliminary  work,  construct  the  circle  diagram  with  this 
degree  of  accuracy.  It  will  be  seen  in  the  course  of  this  treatise 
that  the  efficiency  and  power-factor  can  be  closely  predetermined 
in  spite  of  this  degree  of  indeterminateness  in  the  circle  ratio. 

At  the  author's  request,  Mr.  F.  H.  Kierstead  has  recently 
analyzed  the  test  results  of  130  polyphase  induction  motors, 
and  from  these  results  he  has  derived  a  formula  for  estimating 
the  circle  ratio.  The  range  of  dimensions  of  the  130  motors  may 
be  seen  from  the  following : 

A  varies  from    0.64  mm.  to    2.54  mm. 
Xgr  10       cm.    to  61       cm. 

T  11       cm.    to  84       cm. 

H  7  to  32 

D  20       cm.    to  310     cm. 

Fifty-eight  of  the  motors  were  of  American  manufacture  and 
the  remaining  72  were  of  British,  German,  Swedish,  Swiss,  French, 
and  Belgian  manufacture.  Our  object  was  to  obtain  a  formula 
which  would  yield  an  approximately  representative  value,  regard- 
less of  the  detail  peculiarities  of  design  inherent  to  the  independent 
views  and  methods  of  individual  designers  and  manufacturers. 

For  squirrel-cage  motors  Kierstead's  formula  is  as  follows: 

'0.20  .  0.48  ,  3.0N 

c  =  i 

C  is  a  function  of  A,  and  may  be  obtained  from  Table  28. 
TABLE  28. — VALUES  OF  C  IN  KIERSTEAD'S  FORMULA  FOR  THE  CfRCLE  RATIO. 


A 

(in  mm.). 

C. 

A 
(in  mm.). 

C. 

0.60 

0.71 

'1.60 

1.25 

0.80 

0.88 

1.80 

1.31 

1.00 

0.99 

2.00 

1.39 

1.20 

1.09 

2.20 

1.45 

1.40 

1.17 

2.40 

1.52 

POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   153 

Of  course  it  is  realized  that  the  width  and  depth  of  the  slot, 
width  of  the  slot  openings,  the  arrangement  of  the  end  con- 
nections, the  degree  of  saturation,  and  other  factors  which  have 
not  been  taken  into  consideration,  necessarily  materially  modify 
the  value  of  a,  the  circle  ratio.  Indeed,  all  experienced  designers 
know  that  the  introduction  of  some  extreme  proportion  with 
respect  to  dimensions  additional  to  those  taken  into  account  in 
Kierstead's  formula,  will  be  accompanied  by  quite  a  large  increase 
or  decrease  in  the  circle  ratio.  Consequently,  the  formula  must 
be  taken  as  corresponding  to  average  proportions.  When  any 
extreme  departure  is  made  from  these  average  proportions,  it 
will  be  expedient  for  the  designer  to  employ  his  judgment  as  to 
the  extent  by  which  he  should  modify  the  value  of  the  circle 
ratio  as  obtained  from  the  formula. 

The  investigation  is  at  present  being  continued  with  a  view 
to  arriving  at  a  term  to  be  introduced  into  the  formula  to  take 
into  account  the  slot  dimensions.  In  its  present  form,  however, 
Kierstead's  formula  constitutes  a  valuable  aid  to  the  designer  of 
induction  motors.  Table  27  was  derived  before  Kierstead  under- 
took his  investigation  and  is  based  upon  data  of  71  out  of  the 
130  motors  analyzed  by  Kierstead. 

Below  is  given  a  bibliography  of  a  number  of  useful  articles 
relating  to  the  estimation  of  the  circle-ratio. 

I.  Chapter  IV  (p.  29)  of  Behrend's    "  Induction  Motor." 
II.  "The  Magnetic  Dispersion  in  Induction  Motors,"  by  Dr.  Hans  Behn- 
Eschenburg;  Journal  Inst.  Elec.  Engrs.,  Vol.  32,  (1904),  pp.  239  to  294. 

III.  Chapter  XXI  on  p.  470  of  the  2d  Edition  of  Hobart's  " Electric  Motors." 

IV.  "The  Leakage  Reactance  of  Induction  Motors,"  by  A.  S.  McAllister, 

Elec.  World  for  Jan.  26,  1907. 

V.  "The  Design  of  Induction  Motors,"  by  Prof.  Comfort  A.  Adams,  Trans- 
actions Amer.  Inst.  Elec.  Engrs.,  Vol.  24  (1905),  pp.  649  to  687. 
VI.  "The  Leakage  Factor  of  Induction  Motors,"  by  H.  Baker  and  J.  T. 
Irwin,  Journal  Inst.  Elec.  Engrs.,  Vol.  38,  (1907)  pp.  190  to  208. 

y  and  a  the  Two  Most  Characteristic  Properties  of  a  Design. 
We  have  now  determined  the  two  most  important  characteristics 
of  our  design.  These  are: 

y,  the  ratio  of  the  no-load  current  to  the  full-load  current; 

a,  the  circle  ratio. 

For  our  200-h.p.  design  we  have: 

Y  =  0.242; 

a  =  0,041. 


154       POLYPHASE  GENERATORS  AND  MOTORS 

The  Circle  Diagram  of  the  6-pole  200-H.P.  Motor.    For  the 

full-load  current  we  have: 

7  =  104  amperes. 

No-load  current  =  yX/  =  0.242X104  =  25.2  amperes. 
This  value  of  25.2  amperes  is  plotted  as  AB  in  Fig.  86. 

AB     25/2 

Diameter  of  circle  =  —  =777:17  =  615  amperes. 
a       0.041 

In  Fig.  86,  the  diameter  of  the  circle  is  made  equal  to  615 
amperes. 

Significance  also  attaches  to  the  length  AD  which,  in  Fig. 
86  equals  (25.2+615  =  )  640  amp.  AD  represents  a  quantity 


W 


Wattless  Component  of  Current 

FIG.  86.— The  Circle  Diagram  of  the  200-H.P.  Motor. 

which  may  be  termed  the  "  ideal  short-circuit  current."  It  is 
that  current  which,  if  the  stator  and  rotor  windings  were  of  zero 
resistance  and  if  there  were  no  core  loss,  would  be  absorbed  by 
each  phase  of  the  stator  windings  if  the  normal  pressure  of  1000 
volts  (577  volts  per  phase)  were  maintained  at  the  terminals 
of  the  motor,  the  periodicity  being  maintained  at  25  cycles  per 
second.  In  other  words,  the  reactance,  S,  of  the  motor,  under 
these  conditions,  is: 


=        =  0.902  ohm  per  phase. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   155 

Estimation  of  the  Full-load  Power-factor.  It  will  be  remem- 
bered that  up  to  this  point  we  have  employed  for  the  full-load 
power-factor,  a  preliminary  value  taken  from  Table  18.  This 
preliminary  value  was  Gr  =  0.91. 

In  Fig.  86  the  line  AE  has  been  drawn  with  a  length  of  104 
amperes,  from  A  to  the  point  E  of  intersection  with  the  circle. 
EF,  the  vertical  projection  of  this  line,  is  found  to  be  97  amperes. 

Thus  the  diagramatically-obtained  value  of  the  power-factor  is 


Let  us  take  it  as  0.93  and  let  us  revise  our  data  in  accordance 
with  this  new  value.  We  have: 

Full-load  efficiency  (YJ)         =  0.91  (original  assumption)  ; 
Full-load  power-factor  (G)  =  0.93  (revised  value)  ; 

0  91 

Full-load  current  (I)  =  ^  X  104  =  102  amperes; 

u.y«j 

No-load  current  =25.2  amperes; 

25  2 

X  =  -r~  =  0.247  (revised  value)  ; 
i(jz 

a  =  0.041  (unchanged). 

The  diagram,  drawn  to  a  larger  scale,  and  with  the  slight 
modification  of  employing  this  new  value  of  102  amperes  for  I 
(the  full-load  current),  is  drawn  in  Fig.  87.  It  is  seen  that  the 
vertical  projection  of  I  is  now  95.  This  gives  us  for  the  power- 
factor  : 


confirming  the  readjusted  value. 

The  Stator  I2R  Loss.  We  shall  estimate  the  stator  I2R  loss 
by  first  estimating  the  mean  length  of  one  turn  of  the  winding, 
and  from  this  obtaining  the  total  length  of  conductor  per  phase. 
From  this  length  and  the  already-adopted  cross-section  of  the 


156       POLYPHASE  GENERATORS  AND  MOTORS 

conductor,  the  resistance  per  phase  is  obtained.  We  may  denote 
the  resistance  per  phase  by  R.  The  stator  PR  loss  at  rated 
load  is  equal  to  3I2R. 


^w 

180 

/ 

170 
|160 
gl50 

*% 

o  120 

g  110 
g  100 
a  90 
1    80 
U    70 
t«   60 
2    50 
W    40 
30 
20 
10 

i 

/ 

\\ 

x 

x. 

\ 

I 

-c 

'* 

1 

w 

°< 

1 

V 

1 

X 

^ 

i 

to 

i 

** 

> 

/ 

^o 

1f>r 

/ 

\ 

/ 

\ 

Sty 

aa 

\ 

/ 

<y> 

\ 

\ 

/ 

X 

/ 

\_ 

Wattless  Component  of  Current 

FIG.  87.—  Revised  Circle  Diagram  of  200  H.P.  Motor. 

The  mean  length  of  turn  (mlt.)  may  be  roughly  estimated 
from  the  formula: 


K  is  a  function  of  the  pressure  for  which  the  stator  is  wound, 
and  may  be  taken  from  Table  29. 

TABLE  29.  —  VALUES  OF  K  IN  FORMULA  FOR  MLT. 


Terminal  Pressure  for 
which  Stator  is  Wound. 

K. 

500  (or  less) 

2.5 

1000 

3.0 

2000 

3.5 

4000 

4.0 

6000 

4.5 

8000 

5.0 

10000 

5.5 

12000 

6.0 

POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  157 
For  our  1000-volt  200  h.p.  design,  we  find  the  value: 

#  =  3.0. 

Therefore  : 

mlt.  =  2X43.0+3,OX32.5  =  183.5  cm. 
For  the  number  of  stator  turns  in  series  per  phase  we  have: 

T=120 
Consequently  the  length  of  conductor  per  phase  equals: 

120X183.5  =  22000  cm. 

The  cross-section  of  the  conductor  has  already  been  fixed  at: 
0.302  sq.cm. 

The  PR  loss  at  60°  Cent,  is  desired.  At  this  temperature, 
the  specific  resistance  of  commercial  copper  wire  may  be  taken 
at  0.00000200  ohm  per  cm.  cube. 

Thus  at  60°  Cent,  the  resistance  of  each  phase  of  the  stator 
winding  is: 

22000X0.0000020 


For  the  stator  I2R  loss  at  full  load  we  have: 
3  X1022X  0.146  =  4540  watts. 

Stator  IR  Drop  at  Full  Load.     Instead  of  a  flux  corresponding 

/1000    \ 
to  (  —=-  =     577  volts  per  phase,  we  shall,  at  full  load,  have  a 


lesser  flux.     It  will,  in  fact,  correspond  to  : 

577-102X0.146  =  577-15  =  562  volts. 


158       POLYPHASE  GENERATORS  AND  MOTORS 
This  is  an  internal  drop  of: 

~X  100  =  2.6  per  cent. 

Oil 

Strictly  speaking,  we  ought,  therefore,  to. take  into  account 
the  decreased  magnetic  densities  with  increasing  load.  Cases 
arise  where  it  would  be  of  importance  to  do  this,  but  in  the  present 
instance  such  a  refinement  would  be  devoid  of  practical  inter- 
est and  will  not  be  undertaken. 


THE  DETERMINATION  OF  THE  CORE  LOSS. 

For  the  stator  core,  the  best  low-loss  sheet-steel  should  be 
employed  notwithstanding  that  its  cost  is  still  rather  high.  The 
advantage  in  improved  performance  will  much  outweigh  the  very 
slight  increase  which  its  use  occasions  in  the  Total  Works  Cost. 
In  the  rotor,  the  reversals  of  magnetism  are,  during  normal 
running,  at  so  low  a  rate  that  the  rotor  core  loss  is  of  but  slight 
moment.  It  is  consequently  legitimate  to  employ  a  cheaper 
grade  of  material  in  the  construction  of  the  rotor  cores.  But  in 
practice  it  is  usually  more  economical  to  use  the  same  grade  as 
for  the  stator  cores  notwithstanding  the  absence  of  need  for  the 
better  quality.  By  the  time  the  outlay  for  waste  and  the  outlay 
for  wages  and  for  general  expenses  are  added,  there  will  be  but 
trifling  difference  in  the  cost  of  the  two  qualities. 

The  data  given  in  Table  30  are  well  on  the  safe  side.  Individ- 
ual designers  will  ascertain  by  experience  in  how  far  they  can 
rely  upon  obtaining  better  material. 

TABLE  30. — DATA  FOR  ESTIMATING  THE  CORE  Loss  IN  INDUCTION  MOTORS. 


Density  in  Stator 
Core  in  Lines  per 
Square  Centimeter. 

Core  Loss  in  Stator  Core  in  Watts  per  kg.  for  Various 
Periodicities. 

~=15 

~=25 

~=50 

6000 
8000 
10000 

1.1 

1.7 
2.2 

2.2 
3.0 
4.0 

5.0 

7.4 
10.0 

12000 
14000 

2.6 
3.2 

5.2 
6.2 

POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  159 

For  our  25-cycle  motor,  we  have  employed  in  the  stator  core, 
a  density  of  10  000  lines  per  sq.cm.  and  we  shall  consequently 
estimate  the  core  loss  on  the  basis  of  4.0  watts  per  kg.  of  total 
weight  of  stator  core. 

ESTIMATION  OF  WEIGHT  OF  STATOK  CORE 

External  diameter  of  stator  core  =  89. 2  cm. 
Internal  diameter  of  stator  core  =62.0  cm. 

The  gross  area  of  a  stator  core  plate,  i.e.,  the  area  before  deduct- 
ing the  area  of  the  slots,  is  equal  to : 

|(89.22-62.02)=3260  sq.cm. 

Area  of  72  stator  slots  =  72X5.4X1. 26  =  490  sq.cm. 

Net  area  of  stator  core  plate  =(3260 -490  =  )  2770  sq.cm  . 
The  volume  of  the  stator  core  is  obtained  by  multiplying   this 
area  by  \n,  i.e.,  by  27.9. 

Volume  of  sheet  steel  in  stator  core  =  2770X27.9  =  77  400  cu.cm. 
Weight  of  1  cu.cm.  of  sheet  steel  =  7.8  grams. 

Therefore : 

Weight  of  sheet  steel  in  stator  core  =  —  -  =  603  kg. 

J.UUU 

The  accuracy  with  which  core  losses  can  be  estimated  is  not 
such  as  to  justify  dealing  separately  with  the  teeth  and  the  main 
body  of  the  stator  core.  It  suffices  simply  to  multiply  the  net 
weight  in  kg.  by  the  loss  in  watts  per  kg.  corresponding  to  the 
density  in  the  main  body  of  the  stator  core.  Therefore: 

Stator  core  loss  =  603X4.0  =  2410  watts. 

Core  Loss  in  Rotor.  The  periodicity  of  reversal  of  magneti- 
zation in  the  rotor  core  is  so  low  that  there  should  not  be  much 
core  loss  in  the  rotor.  But  to  allow  for  minor  phenomena  and  to 
keep  on  the  safe  side,  it  is  a  good  rule  to  assess  the  rotor  core 


160       POLYPHASE  GENERATORS  AND  MOTORS 

loss  at  10  per  cent  of  the  stator  core  loss.     For  our  machine  we 
have : 

.      Rotor  core  loss  =  2410  X  0.10  =  240  watts. 

Input  to  Motor  and  to  Rotor  at  Rated  Load.  We  cannot 
yet  check  our  preliminary  assumption  of  an  efficiency  of  91  per 
cent  at  rated  load.  On  the  basis  of  this  efficiency,  the  input  to 
the  motor  at  its  rated  load  is: 

200X746 

—  =164  000  watts. 
L/.y  -L 

The  losses  in  the  stator  amount  to  a  total  of : 
4540+2410  =  6950  watts. 

Deducting  the  stator  losses  at  full  load  from  the  input  to  the 
motor  at  full  load  we  ascertain  that: 

164  000  -  6950  =  157  050  watts 

are  transmitted  to  the  rotor. 

A  Motor  is  a  Transformer  of  Energy.  A  motor  receives 
energy  in  the  form  known  as  "  electricity."  An  account  can  be 
rendered  of  all  the  energy  received.  In  the  case  we  are  consider- 
ing, the  full-load  input  is  at  the  rate  of  164  000  watts.  Energy 
flows  into  the  motor  at  the  rate  of  164  kw.  hr.  per  hour.*  In  cer- 
tain instances  it  is  more  convenient  to  make  some  equivalent  state- 
ment with  other  units  of  power  and  time.  Thus  we  may  say  that 
energy  flows  into  the  motor  at  the  rate  of  164  000  watt  seconds 
per  second.  The  amount  of  energy  corresponding  to  the  expendi- 
ture of  one  watt  for  one  second  is  termed  by  the  physicist,  one 
joule. 

1  joule  =  1  watt  second. 

It  requires  4190  joules  to  raise  the  temperature  of  I  kg.  of 
water  by  1°  cent.  Thus  we  have: 

1  kg.  calorie  (kg.cal.)  =4190  joules. 

*  The  proposal  to  designate  as  1  kelvin  the  amount  of  energy  correspond- 
ing to  1  kw.  hr.,  is  gradually  gaining  favor  amongst  European  engineers. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  161 

The  energy  expended  in  lifting  a  weight  of  1  kg.  through  a 
height  of  1  meter  is  equivalent  to  9.81  joules.  In  other  words 
1  kg.m.  =  9.81  joules. 

The  energy  need  not  necessarily  be  expended  in  lifting  a  weight. 
If  a  force  of  1  kg.  is  exerted  through  a  distance  of  1  meter,  an 
amount  of  energy  equal  to  9.81  joules  is  expended  in  the  process. 

The  amount  of  energy  corresponding  to  the  expenditure  of 
1  h.p.  for  1  sec.  is  equal  to  746  joules;  or 

1  h.p.  sec.  =  746  joules. 
It  is  convenient  to  bring  together  these  equivalents: 

1  watt  second  =  1  joule; 
1  h.p.  second  =746  joules; 
1  kg.m.  =9.81  joules; 

1  kg.cal.  =4190  joules. 

1  kelvin  =  3  600  000  joules. 

The  rated  output  of  our  motor  may  be  expressed  as: 

1.  200  h.p.; 

la.    200  h.p.  sec.  per  sec.; 

2.  149  200  watts; 

2a.    149  200  joules  per  second. 


3.  =     15  2°0  kg.m.  per  second. 


/149  200     \  . 

\   4190    =  /  kg.cal.  per  second. 

Let  us  concentrate  our  attention  on  Designation  3  for  the 
rated  output.  In  accordance  with  this  designation,  the  motor's 
output  at  its  rated  load  is 

15  200  kg.m.  per  sec. 

If  the  shaft  of  our  motpr  is  supplied  with  a  gear  wheel  of  1 
meter  radius  through  which  it  transmits  the  energy  to  another 
engaging  gear  wheel,  and  thence  to  the  driven  machinery,  then 


162       POLYPHASE  GENERATORS  AND  MOTORS 

for  every  revolution  of  the  armature,  the  distance  travelled  by  a 
point  on  the  periphery  of  the  gear  wheel  is 

2Xx  =  6.28  meters. 

At  the  motor's  synchronous  speed  of  500  r.p.m.,  the  peripheral 
speed  of  the  gear  wheel  is 

500 
6.28  X-gQ-  =  52.3  meters  per  sec. 

(Although  such  a  high  speed  would  not,  in  practice,  be 
employed,  it  has  been  preferable,  for  the  purpose  of  the  present 
discussion,  to  consider  a  gear  wheel  of  1  meter  radius.) 

Since  at  full  load  the  motor's  output  is  : 

15  200  kg.m.  per  sec. 

and  since  the  peripheral  speed  of  the  gear  wheel  is  52.3  meters 
per  sec.,  it  follows  that  the  pressure  at  the  point  of  contact  between 
the  driving  and  the  driven  gear  teeth  is 


This  is  the  force  exerted  at  a  radius  of  1  meter.  We  say 
that  at  full  load  the  motor  exerts  a  "  torque  "  of 

290X1  =  290  kg.  at  1  meter  leverage. 

Had  the  radius  of  the  gear  been  only  0.5  m.  instead  of  1.0 
meter,  then  the  force  would  have  been  2X290  =  580  kg.,  but  the 
"  torque  "  would  still  have  been  equivalent  to 

0.5X580  =  290  kg.  at  1  meter  leverage. 

In  dealing  with  torque  it  is  usually  convenient  to  reduce  it 
to  terms  of  the  force  in  kg.  at  1  meter  leverage,  irrespective  of 
the  actual  leverage  of  the  point  of  application  of  the  force. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  163 

At  rated  load  our  motor  will  not  run  at  the  synchronous  speed 
of  500  r.p.m.,  but  at  some  slightly  lower  speed.  If  the  "  slip  " 
is  1  per  cent,  then  the  full-load  speed  will  be 

0.99X500  =  495  r.p.m. 

If  the  "  slip  "  is  2  per  cent,  then  the  full-load  speed  will  be 
0.98X500  =  490  r.p.m. 

Since  the  rated  output  will  nevertheless  be  200  h.p.  in  these 
two  cases,  the  torque  will  be;  not  290  kg.  at  1  m.  leverage,  but 


and 


/  290 

(  rTnn  =  )  293  kS-  *  °'r  tne  case  °*  *  Per 

u.yy 


(290 
—  —  =  1  296  kg.  for  the  case  of  2  per  cent  slip. 
\J  •  i/O 


The  torque  (i.e.,  the  force  at  1  m.  leverage)  is  only  one  com- 
ponent of  the  energy  delivered.  The  other  component  is  the 
distance  traversed  by  a  hypothetical  point  at  1  meter  radius 
revolving  at  the  angular  speed  of  the  rotor.  For  2  per  cent  slip, 
the  distance  traversed  in  1  sec.  by  such  a  hypothetical  point,  is 

0.98X52.4  =  51.  3  meters. 
The  energy  delivered  from  the  motor  in  each  second  is  thus 

296X51.3  =  15200kg.m. 
The  power  (or  rate  of  deliverance  of  energy)   is 

15  200  kg.m.  per  sec.; 
or, 

9.81  X  15  200  =  149  200  watts; 


or, 

149  200 
746 


=  200  h.p. 


164       POLYPHASE  GENERATORS  AND  MOTORS 

The  torque  exerted  by  the  rotor  conductors  is  greater  than 
that  finally  available  at  the  gear  teeth.  The  discrepancy  cor- 
responds to  the  amount  of  the  PR  loss  in  the  rotor  conductors 
and  to  the  amount  of  the  rotor  core  loss  and  the  windage  and 
bearing  friction.  The  rotor  core  loss  and  the  friction  come 
in  just  the  same  category  as  an  equal  amount  of  external 
load.  Thus  if  3  h.p.  is  required  to  supply  the  rotor  core 
loss  plus  friction,  then  the  output  from  the  rotor  conductors 
is  203  h.p.  as  against  the  ultimate  output  of  200  h.p.  from 
the  motor. 

But  the  PR  loss  in  the  rotor  conductors  comes  in  an  altogether 
different  category.  The  loss  can  only  come  about  as  the  result  of  a 
cutting  of  the  flux  across  the  rotor  conductors.  In  other  words, 
the  rotor  conductors  must  not  travel  quite  as  fast  as  the  revolving 
magnetic  field.  Consequently  the  rotor  will  run  at  a  speed 
slightly  less  than  synchronous;  there  will  be  a  "  slip  "  between 
the  revolving  magnetic  field  and  the  revolving  rotor.  It  is  only 
in  virtue  of  such  slip  that  the  rotor  conductors  can  be  the  seat 
of  any  force.  Thus  the  torque  is  inseparably  associated  with 
the  "  slip  "  and  the  "  slip  "  will  be  greater  the  greater  the  load. 
As  the  "  slip  "  and  torque  increase,  the  rotor  PR  loss  also  increases 
and  the  speed  of  the  rotor  decreases. 

If  the  PR  loss  in  the  rotor  conductors  amounts  to  1  per  cent 
of  the  input  to  the  rotor,  then  the  "  slip  "  will  be  1  per  cent. 
If  the  PR  loss  is  increased  to  2  per  cent,  then  the  "  slip  "  increases 
to  2  per  cent.  If,  finally,  the  PR  loss  amounts  to  100  per  cent  of 
the  input  to  the  rotor,  then  the  "  slip  "  will  be  100  per  cent, 
i.e.,  the  motor  will  be  at  rest,  but  it  may  nevertheless  be  exerting 
torque.  For  such  a  condition  it  is  desirable  to  regard  matters 
from  the  following  standpoint.  If  the  rotor  is  suitably  secured 
so  that  it  cannot  rotate;  then  if  electricity  is  sent  into  the  motor 
a  certain  portion  will  be  transmitted  by  induction  to  the  rotor 
circuit,  just  as  if  it  constituted  the  secondary  circuit  of  a  trans- 
former. The  input  to  the  rotor  will  under  these  conditions 
consist  of  the  PR  loss  in  its  conductors  and  the  core  loss.  There 
is  no  other  outlet  for  the  energy  sent  into  the  rotor  and  it  all 
becomes  transformed  into  energy  in  the  form  of  heat  in  the  rotor 
conductors  and  in  the  rotor  core.  Since  under  such  conditions 
the  rotor  core  loss  is  negligible  in  comparison  with  the  rotor 
PR  loss,  we  may  regard  the  input  to  the  rotor  as  practically 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   165 


identical  with  the   rotor  PR  loss.      The  rotor  PR  loss  is  100 
per  cent  of  the  input  to  the  rotor  and  the  slip  is  100  per  cent. 

The  Locus  of  the  Rotor  Current  in  the  Circle  Diagram.  Let 
us  consider  the  hypothetical  case  of  a  rotor  with  a  number  of 
conductors  equal  to  the  number  of  stator  conductors.  ,  We  can 
describe  this  as  an  arrangement  with  a  1 : 1  ratio  of  transformation. 
Any  actual  induction  motor  can  be  considered  to  have  its  equivalent 
with  a  1 : 1  ratio,  and  it  is  convenient  and  usual  to  investigate 
certain  properties  of  induction  motors 
by  considering  the  equivalent  rotor  wind- 
ing with  a  1 : 1  ratio. 

The  vector  diagram  of  the  stator  and 
rotor  currents  is  shown  in  Fig.  88  for 
our  200-h.p.  motor  with  a  l:l-ratio 
rotor.  The  diagram  is  drawn  for  full 
load  and  consequently  the  stator  current, 
AE,  is  equal  to  102  amperes.  The  rotor 
current,  AF  must  have  such  direction 
and  magnitude  that  the  resultant,  AB, 
shall  be  equal  to  the  no-load  magnetizing 
current,  which  we  have  already  found  to 
be  25.2  amperes.  AF  is  consequently 
equal,  as  regards  phase  and  magnitude, 
to  EB  and  is  found  graphically  to  amount 
to  96  amperes.  In  practice,  it  is  more 
convenient  to  represent  the  rotor  current 
by  lines  drawn  from  B  as  the  origin  and 
connecting  B  to  the  points  where  the 
corresponding  primary  vectors  intersect 
the  circumference  of  the  semi-circle.  In 
Fig.  89  are  drawn  the  stator  and  rotor 
vectors  for  two  values  of  the  stator  cur- 
rent, namely  80  amperes  and  300  amperes. 

The  corresponding  values  of  the  rotor  current  are  72  and  286 
amperes. 

The  Rotor  I2R  Loss  of  the  200-H.P.  Motor  at  Its  Rated  Load. 
It  is  still  desirable  to  postpone  to  a  later  stage  the  design  of  the 
rotor  conductors.  But  let  us  assume  that  the  full-load  slip 
will  be  2.0  per  cent.  Then  the  rotor  PR  loss  at  full  load  will 
be  2.0  per  cent  of  the  input  to  the  rotor.  The  input  to  the  rotor 


FIG.  88. — Vector  Diagram 
Indicating  the  Primary 
and  Secondary  Currents 
in  the  200-H.P.  Induc- 
tion Motor  at  its  Rated 
Load. 


166       POLYPHASE  GENERATORS  AND  MOTORS 


has  been  ascertained  (on  page  160)  to  be  157  050  watts.      Thus 
the  rotor  PR  loss  at  full  load  is  0.02X157  050  =  3140  watts. 

We  have  now  determined  (or  assumed)  all  the  full-load  losses 
except  windage  and  bearing  friction. 


/ 


// 


20     40     60    80    100  120  140  160  180  200  220  240  260  280   300  320   340 
Wattless  Component  of  Current 

FIG.  89. — Diagram  Indicating  Stator  and  Rotor  Current  Vectors. 

Friction  Losses.  It  is  very  hard  to  generalize  as  regards 
friction  losses.  A  reasonable  estimate  may,  however,  be  made 
from  Table  31. 

TABLE  31. — DATA  FOR  ESTIMATING  THE  FRICTION  Loss  IN  BEARINGS  AND 
WINDAGE  IN  INDUCTION  MOTORS. 


Z>2X0  (D  and  \g  in  dm.) 

Total  Friction  Loss  in  Watts,  for  Various  Rated  Speeds. 

400  r.p.m. 

800  r.p.m. 

1600  r.p.m. 

10 
20 

30 

150 
280 
370 

340 
620 
800 

1200 

2000 
2500 

40 
50 
100 

460 
550 
800 

1000 
1250 
1750 

3100 
3700 
5600 

200 

1100 

2400 

8000 

POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   167 
For  our  motor  we  have  : 


(i  £40\ 
=  ^M=2.06h.p. 


=  166. 

From  Table  31  we  ascertain  by  interpolation  that  the  friction 
loss  may  be  taken  as  1300  watts. 

Output  from  Rotor  Conductors.  The  output  from  the  rotor 
conductors  is  made  up  of  the  200-h.p.  output  from  the  motor, 
the  friction,  and  the  rotor  core  loss.  The  two  latter  amount  to 

Thus  we  may  take  the  output  from  the  rotor  conductors  as 
200+2.06  =  202  h.p. 

Referring  back  to  page  163  we  find  that  for  2  per  cent  slip 
and  200  h.p.,  we  require  296  kg.  at  1  meter  leverage. 

Consequently  the  torque  required  to  be  exerted  by  the  rotor 
conductors  is  : 

202 

—  -X296  =  299  kg.  at  1  meter  leverage. 


In  other  words,  the  full-load  torque  exerted  by  the  rotor  conductors 
is  299  kg. 

The  Torque  Factor.  At  full  load,  the  input  to  the  rotor  is 
157  050  watts.  Thus  the  input  to  the  rotor  in  watts  per  kg. 
of  torque  developed,  which  we  may  term  the  "  torque  factor,"  is: 

157050__0. 
~299~ 

This  factor  will  be  useful  to  us  in  studying  the  starting  torque. 

The  "  Equivalent  "  Resistance  of  the  Rotor.  We  have  on 
p.  157  made  an  estimation  of  the  stator  resistance  and  have 
ascertained  it  to  be  0.146  ohm  per  phase  at  60°  cent. 

The  stator  PR  loss  at  full  load  is  3  X1022X  0.146  =  4540  watts. 

The  rotor  I2R  loss  at  full  load  is  3140  watts. 


168       POLYPHASE  GENERATORS  AND  MOTORS 

We  shall  employ  a  squirrel-cage  rotor  (for  which  we  shall 
soon  design  the  conductors)  and  we  may  consider  its  "  equivalent  " 
resistance  to  be:. 


X  0.146  =  0.101  ohm. 


Without  introducing  any  serious  inaccuracy  we  may  (when 
the  motor  is  running  in  the  neighborhood  of  synchronous  speed), 
ascertain  the  rotor  PR  loss  for  any  value  of  the  stator  current 
(except  for  very  small  loads)  by  multiplying  this  "  equivalent  " 
resistance  by  three  times  the  square  of  the  stator  current. 

Rotor  at  Rest.  But  when  the  rotor  is  at  rest,  the  currents 
circulating  in  its  windings  are  of  the  line  periodicity  and  the 
conductors  have  an  apparent  resistance  materially  greater  than 
their  true  resistance.  Attention  was  called  to  a  related  phe- 
nomenon in  a  paper  presented  by  A.  B.  Field,  in  June,  1905,  before 
the  American  Institute  of  Electrical  Engineers  and  entitled 
"  Eddy  Currents  in  Large  Slot- Wound  Conductors."*  Recently 
the  application  of  the  principle  has  been  incorporated  in  the 
design  of  squirrel-cage  induction  motors  to  endow  them  with 
desired  values  of  starting  torque.  The  multiplier  by  which  the 
apparent  resistance  may  be  obtained  from  the  true  resistance  may 
be  found  approximately  from  the  formula: 


Multiplier  =  0. 15  X  (depth  of  rotor  bar  in  cm.)  X  Vperiodicity. 
In  our  case  we  have: 

Multiplier  =  0. 15  X  5.4  X  V25  =  4.05. 

This  multiplier  only  relates  to  the  embedded  portions  of  the 
conductors.  The  portions  of  the  length  where  the  conductors 
cross  the  ventilating  ducts  are  not  affected,  nor  are  the  end  rings 
subject  to  this  phenomenon.  A  rough  allowance  for  this  can  be 
made  by  reducing  the  multiplier  to  0.8X4.05  =  3.24. 

Thus  at  standstill  the  "  apparent  "  resistance  of  our  rotor  will 
be  3.24X0.101=0.327  ohm. 

"Vol.  24,  p.  761. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  169 

The  "  Equivalent  "  Resistance  of  the  Motor.  At  standstill, 
the  motor,  regarded  as  a  whole,  and  without  distinction  between 
primary  and  secondary  may  be  considered  as  having  a  resistance 
of  (0.146+0.327  =  )  0.473  ohm  per  phase. 

On  page  154  we  have  seen  that  the  reactance  of  the  motor  is: 

S=  0.902  ohm  per  phase. 

For  the  impedance  of  the  motor  at  standstill  we  have: 
Vo.4732+0.902^1.03  ohm  per  phase. 

THE  STARTING  TORQUE 

In  starting  this  motor  we  should  not  put  it  at  once  across 
the  full  pressure  of  1000  volts  but  should  apply  only  one-half  or 
one-third  of  this  pressure.  Let  us  examine  the  conditions  at 
standstill  when  half  pressure  is  applied. 

We  shall  have  —^  =  288  volts  per  phase. 
The  stator  current  will  be: 

OQQ 

—  •—  =  280  amperes  per  phase. 

For  the  rotor  PR  loss  we  have  3X2802X0.327  =  77  000  watts. 
The  torque  developed  is  obtained  by  dividing  this  value  by 
the  torque  factor,  i.e.,  by  525. 

77  000 
Torque  =    _0      =  146  kg.  at  1  meter  leverage. 


The  full-load  torque  is  296  kg. 

Consequently  with  half  the  normal  pressure  at  the  motor,  we 
shall  obtain 

146 

49.5  per  cent  of  full-load  torque. 


170       POLYPHASE  GENERATORS  AND  MOTORS 

We  obtain  the  half  pressure  at  the  motor  by  tapping  off  from 
the  middle  point  of  a  starting  compensator.  The  connections 
(for  a  quarter-phase  motor),  are  as  shown  in  Fig.  90.  With  this 
arrangement,  the  current  drawn  from  the  line  will  be  only  half  of 
the  current  taken  by  the  motor.  Since  the  motor  takes  280 
amperes,  the  current  from  the  line  is  only 

/280     \  , 

I  - s-=  )  140  amperes. 

(140     \ 
TQO  =  )  1-37  times  full-load  current  from  the 

line,  we  can  start  the  motor  with  50  per  cent  of  full -load  torque. 


FIG.  90. — Connections  for  Starting  Up  an  Induction  Motor  by  Means  of  a 
Compensator  (sometimes  called  an  auto-transformer). 


This  excellent  result  is  achieved  by  employing  deep  rotor  con- 
ductors and  does  not  involve  the  necessity  of  resorting  to  high 
slip  during  normal  running. 

Circle  Diagram  for  These  Starting  Conditions.  For  half 
pressure,  the  magnetizing  current  will,  strictly  speaking,  be  a 
little  less  than  half  its  former  value  of  25.2  amperes,  since  the 
magnetic  parts  are  worked  at  lower  saturation.  But  for  simplicity 

(25  2     \ 
— ~  =  }  12.6    amperes.     The    circle  ratio,  <j, 

varies  a  little  with  the  saturation,  but  let  us  take  it  at  its  former 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  171 

value  of  0.041.     We  can  now  construct  the  circle  diagram  shown 
in  Fig.  91,  in  which: 

AB  —  12.6  amperes; 


=        amperes; 

AD  =  12.6+308  =  321; 
AE  =  280  amperes. 


16U 
150 

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§  130 

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V 

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1020304050607080     100     120     140     160     180     200     220     240\  260     280    300     320 
Wattless  Component  of  Current 

FIG.  91. — Starting  Torque    Diagram  for  200-H.P.  Motor  when  Connected 
for  Starting  from  Half  the  Normal  Line  Pressure. 

A  vertical  line  EF  is  drawn  from  E  to  the  base  line.  This 
is  the  vertical  projection  of  280  amperes  and  represents  the  energy 
component  of  the  input  to  the  motor  under  these  conditions  of 
standstill.  From  the  diagram  we  ascertain  graphically  that: 

#^=131  amperes. 
Consequently  the  input  to  the  motor  is: 

=  113000  watts. 


The  stator  PR  loss  =  3X2802X0.146  =  34  300  watts. 
Subtracting  this  from  the  input,  we  ought  to  obtain  the  PR 
loss  in  the  rotor. 

Rotor  PR  loss  =  113  000-34  300  =  78  700  watts. 


172       POLYPHASE  GENERATORS  AND  MOTORS 

This  is  in  good  agreement  with  the  value  of  77  000  watts  which 
we  obtained  for  the  rotor  PR  loss  by  applying  the  analytical 
method. 

EF  is,  for  constant  pressure,  a  measure  of  the  input  to  the  motor 
and  is  to  the  scale  of 

113  000 

—  0       =  865  watts  per  ampere. 
lol 

[It  can  also  be  seen  that  this  would  be  the  case  from  the  cir- 
cumstance that  at  a  pressure  of  288  volts  per  phase,  the  input 
is  equal  to : 

(3X288X7)  =  (865  7)  watts]. 

The  rotor  PR  loss  of  78  700  watts  may  then  be  represented 
by  the  height  FG  corresponding  to  (  =  )  91.0  amperes. 

\     oDO  / 

and  the  stator  PR  loss  may  be  represented  by  the  remainder, 
GE,  corresponding  to 


=.    amperes. 


It  also  necessarily  follows  that  FG  (and  corresponding  vertical 
heights  for  other  conditions  similarly  worked  out),  is  a  measure 
of  the  torque.  When  used  in  this  way,  the  scale  is 

146 

j-r-  r  =  1.60  kg.  (at  1  meter  leverage)  per  amp. 


Some  General  Observations  Regarding  the  Circle  Diagram. 

It  is  this  adaptability  to  the  forming  of  mental  pictures  of  the 
occurrences,  which  renders  the  circle  diagram  of  great  importance 
in  the  design  of  induction  motors.  All  the  various  calculations 
involved  in  induction-motor  design  may  be  carried  through  by 
analytical  methods  but  it  is  believed  that  these  exclusively  analyt- 
ical methods  are  inferior  in  that  they  disclose  no  simple  picture 
of  the  occurrences.  It  is  well  known  that  in  practice  the  locus 
of  the  extremity  of  the  stator-current  vectors  is  rarely  more  than 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  173 

a  very  crude  approximation  to  the  arc  of  a  circle  and  that  the 
designer  can  only  consider  as  rough  approximations  the  results 
he  deduces  on  the  basis  of  the  circle  assumption.  But  when 
employed  with  judgment  the  plan  is  of  great  assistance  and  suffices 
to  yield  reasonable  results. 


THE   SQUIRREL   CAGE 

Let  us  now  design  our  rotor's  squirrel  cage.  As  yet  we  have 
merely  prescribed  that  the  full-load  I2R  loss  in  the  squirrel  cage 
shall  be  3140  watts,  that  the  slots  shall  be  54  mm.  deep  and  that 
we  shall  employ  a  number  of  slots  not  differing  very  materially 
from  72,  the  number  of  stator  slots. 

The  Number  of  Rotor  Slots.  Were  we  to  employ  72  rotor 
slots,  the  motor  would  have  a  strong  "  cogging  "  tendency.  That 
is  to  say,  if,  with  the  rotor  at  rest,  pressure  were  applied  to 
the  stator  terminals,  there  would  be  a  strong  tendency  for  the 
rotor  to  "  lock,"  at  a  position  in  which  there  would  be  a  rotor 
slot  directly  opposite  to  each  stator  slot.  This  tendency  would 
very  markedly  interfere  with  the  development  of  the  starting 
torque  calculated  in  a  preceding  section.  The  choice  of  71  or 
73  slots  would  eliminate  this  defect,  but  might  lead  to  an  unbal- 
anced pull,  slightly  decreasing  the  radial  depth  of  the  air-gap  at 
one  point  of  the  periphery  and  correspondingly  increasing  its 
depth  at  the  diametrically  opposite  point.  This  excentricity 
once  established,  the  gap  at  one  side  would  offer  less  magnetic 
reluctance  than  the  gap  diametrically  opposite,  thus  increasing 
the  dead-point  tendency.  But  by  selecting  70  or  74  slots,  this 
defect  is  also  eliminated.  Let  us  determine  upon  70  rotor  slots 
for  our  design. 

It  may  in  general  be  stated  that  the  tendency  to  dead  points 
at  starting  will  be  less. 

1.  The  smaller  the  greatest  common  divisor  of  the  numbers 

of  stator  and  rotor  slots. 

2.  The  greater  the  average  number  of  stator  and  rotor  slots 

per  pole. 

3.  The  less  the  width  of  the  slot  openings. 

4.  The  greater  the  resistance  of  the  squirrel-cage. 

5.  The  deeper  the  rotor  slots. 


174       POLYPHASE  GENERATORS  AND  MOTORS 

The  influence  of  the  last  two  factors  will  be  better  understood 
if  it  is  pointed  out  that  they  determine  the  rotor  PR  loss  at  start- 
ing, and  we  have  already  seen  that  the  starting  torque  is  pro- 
portional to  the  PR  loss  in  the  rotor.  The  fluctuations  in  the 
starting  torque  arising  from  variations  in  the  relative  positions 
of  the  stator  and  rotor  slots,  will  obviously  be  a  smaller  percentage 
of  the  average  starting  torque,  the  greater  the  absolute  value  of 
the  average  starting  torque.  Thus  if  the  average  starting  torque 
is  very  low,  a  small  fluctuation  might  periodically  reduce  it  to 
zero;  i.e.,  the< motor  would  have  dead  points.  If,  on  the  other 
hand,  the  average  starting  torque  is  high,  these  same  fluctuations, 
superposed  on  this  high  average  starting  torque,  would  still  leave 
a  high  value  for  the  minimum  torque,  and  there  would  be  no 
dead  points. 

The  Pitch  of  the  Rotor  Slot.  The  external  diameter  of  the  rotor 
is  (620-2X1.3  =  )  617.4  mm. 

The  diameter  at  the  bottom  of  the  slots  is : 

(617.4 -2X54  =  )  509.4  mm. 

Consequently  the  rotor  slot  pitch  at  the  bottom  of  the  slot  is : 
509.4  Xx 


70 


=  22.8  mm. 


The  depth  of  the  rotor  conductor  can  be  practically  identical 
with  the  depth  of  the  slot.  Therefore  depth  of  rotor  conductor 
=  54  mm. 

Ratio  of  Transformation.  We  have  72  stator  slots  and  10  con- 
ductors per  slot;  hence  a  total  of  720  stator  conductors;  as  against 
only  70  rotor  conductors.  The  ratio  of  transformation  is  thus: 

720  :  70  =  10.3  :  1. 

We  have  already  estimated  that  for  a  1  :  1  ratio,  the  rotor 
current  would,  at  full  load,  amount  to  96  amperes.  We  are  now 
able  to  state  that  with  the  actual  ratio  of  10.3  :  1,  the  current 
in  the  rotor  face  conductors  will  be  10.3X96  =  990  amperes. 

At  each  end  of  the  rotor  core,  the  rotor  conductors  will  termin- 
ate in  end  rings.  It  will  be  desirable,  for  structural  reasons,  to 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  175 

space  these  end  rings  (in  this  particular  motor),  2.0  cm.  away  from 
the  ends  of  the  laminated  core.  Consequently  the  length  of  each 
conductor  between  end  rings  will  amount  to 

43+2X2  =  47  cm. 

The  PR  loss  in  the  70  rotor  conductors  will  be  equal  to  that 
in  a  single  conductor  of  the  same  (as  yet  undetermined)  cross- 
section,  but  with  a  length  of 

70X47  =  3290  cm. 

We  have  seen  that  we  wish,  at  full  load,  to  have  a  loss  of  3140 
watts  in  the  squirrel  cage.  Were  the  loss  in  the  end  rings  negli- 
gible (the  entire  3140  watts  being  dissipated  in  the  70  face  con- 
ductors, then  their  section  would  be  so  chosen  that  they  should 
have  an  aggregate  resistance  of: 


=  0.00320  ohm. 


Since  at  60°  cent.,  the  specific  resistance  of  commercial  copper 
is  0.00000200,  the  required  cross-section  would  be: 

3290X0.00000200 

0.00320  =  2.06sq.cm. 

Since  the  depth  of  a  rotor  conductor  is  54  mm.,  its  width  would 
thus  require  to  be: 

=  3.82  mm.   -W, 


But  we  cannot  afford  to  provide  so  much  material  in  the  end 
rings  as  to  render  them  of  practically  negligible  resistance.  Let 
us  plan  to  allow  a  loss  of  628  watts  (20  per  cent)  in  the  end  rings, 
the  remaining  (3140  —  628  =  )  2512  watts  occurring  in  the  slot  con- 
ductors. This  will  require  an  increase  in  the  width  of  the  slot 
conductors,  to: 


=  4.78  mm.     or    4.8mm. 


176       POLYPHASE  GENERATORS  AND  MOTORS 

Thus  the  slot  conductors  will  have  a  cross-section  of  54 
mm.X4.8  mm.  The  rotor  slot  will  also  be  54  mm.  deep  and 
4.8  mm.  wide,  and  there  will  be  no  insulation  on  the  rotor  con- 
ductors. 

The  End  Rings.  The  object  in  minimizing  the  loss  in  the 
end  rings  will  have  been  divined.  Since  we  want  to  have  the 
apparent  resistance  at  starting,  as  great  as  possible  for  a  given 
"  slip  "  at  normal  load,  we  want  to  concentrate  the  largest  prac- 
ticable portion  of  the  loss  in  the  slot  conductors  since  it  is  these 
conductors  which  manifest  the  phenomenon  of  having,  for  high- 
periodicity  currents,  a  loss  greatly  in  excess  of  that  occurring  when 
they  are  traversed  by  currents  of  the  low  periodicity  corresponding 
to  the  "  slip." 

The  Current  in  the  End  Rings.  It  can  be  shown*  that  the 
current  in  each  end  ring  is  equal  to: 

Number  of  rotor  conductors  , 

—^ — ; —      -  X  current  per  slot  conductor. 
iuX  number  of  poles 

For  our  motor  we  have: 

Full-load  current  in  each  end  ring  =  -—^X  990  =  3660  amperes. 


Since  we  have  allowed  628  watts  for  the  loss  in  the  end  rings; 
we  have  a  loss  of  314  watts  per  end  ring,  and  we  have: 

Resistance  for  one  end  ring  =  -      ^2  =  0.0000234  ohm. 

Each  end  ring  will  have  a  mean  diameter  of: 

61.7+50.9 

^ =  56.3  cm. 

and  a  mean  circumference  of  56.3x  =  177  cm. 

Consequently  the  cross-section  of  each  end  ring  is : 

177X0.00000200  Q 

0.0000234 

*  The  proof  is  given  in  Chapter  XXIII  (pp.  490  to  492)  of  the  2d  Edition 
of  the  author's  "  Electric  Motors  "  (Whittaker  &  Co.,  1910). 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   177 


Let  us  make  the  cross-section  up  with  a  height  of  54  mm. 
and  a  width  of  28  mm.     A  conductor  of  this  cross-section  will,  at 


With 


11.7    teeth 


<-4.8  mm 


— 22  mm 


Section  on  A.B  Looking  in  Direction 
of  Arco.ws    , 

FIG.  92.— Outline  of  Squirrel-cage  Rotor  for  200-H.P.  Polyphase  Induction 

Motor. 

25  cycles,  be  subject  to  a  slight  increase  in  resistance,  due  to 

ordinary  skin  effect,  but  this  will  not  be  of  sufficient  amount  to 

add    much    to  the   starting 

torque.     The  increase  in  re-  — >j  k-2.4mm. 

sistance  will  in  fact  amount 

to  about  10  per  cent. 

A  sketch  of  the  rotor  is 
given  in  Fig.  92,  and  a  section 
of  a  slot  and  two  teeth,  in 
Fig.  93.  Since  the  slot  pitch 
at  the  bottom  of  the  slot  is 
22.8  mm.  and  since  the  slot 
is  4.8  mm.  wide,  we  have  a 
tooth  width  of  22.8-4.8  = 
18.0  mm. 

(?=) 

per  pole,  the  cross-section  of 
the  magnetic  circuit  at  the 
bottom  of  the  rotor  teeth  is 


FIG.  93.— Rotor  Slot  and  Teeth  for  200- 
H.P.  Squirrel-cage  Motor. 


11.7X1.8X27.9  =  588  sq.cm. 


178       POLYPHASE  GENERATORS  AND  MOTORS 

The  crest  density  at  the  bottom  of  the  rotor  teeth  is: 

4  570  000 

X  1.7  =  13  200  lines  per  sq.cm. 


588 


THE  EFFICIENCY 


We  have  now  estimated  all  the  losses  in  our  200-h.p.  motor. 
Let  us  bring  them  together  in  an  orderly  table : 

At  full  load  we  have : 

Stator  PR  loss 4  540  watts 

Stator  core  loss 2410     " 

Rotor  PR  loss 3140     " 

Rotor  core  loss 240     " 

Friction  and  windage  loss 1  300     ' ' 


Total  of  all  losses 11  630  watts 

Output  at  full  load 149  200     ' ' 


Input  at  full  load 160  830  watts 

14.0  onrj 

Full-load  efficiency  =  -       £ X 100  =  93.0  per  cent, 
lou  ooU 

Our  original  assumption  for  the  full-load  efficiency  was  91.0 
per  cent.  Consequently,  strictly  speaking,  we  ought  to  revise 
several  quantities,  such  as  current  input,  stator  PR  loss,  rotor 
PR  loss,  and  ultimately  obtain  a  still  closer  approximation  to 
the  efficiency.  But  the  object  of  working  through  this  example 
has  been  to  convey  information  with  respect  to  the  methods  of 
carrying  out  the  calculations  involved  in  the  design  of  an.  induc- 
tion motor ;  and  there  is  no  special  reason  to  undertake  the  above 
mentioned  revision. 


THE  HALF-LOAD  EFFICIENCY 

At  half  load,  the  energy  component  of  the  full-load  current 
input,  will,  sufficiently  exactly  for  our  purpose,  be  halved.  This 
energy  component  is: 

GXl  =  0.93X102  =  95  amperes. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   179 


At  half  load,  the  wattless  component  will  be  slightly  in  excess 
of  25.2  amperes,  the  no-load  current.  Let  us  take  it  as  29 
amperes.  The  total  current  input  at  half  load  will  be: 


55.5  amperes. 

The  stator  PR  loss  at  half  load  will  be: 

X  4540  =  1340  watts. 


/55.5\2 
U027 


+*  34 

I  32 
^30 

*  28 

4J 

g  26 

I  24 
I  22 

&  2° 
<3  18 

W  16 
14 
12 
10 


7 


Dia. 


308-A 


2     4     6     8    10  12  14  16   18  20  22  24   26  28  30  32  34  36   38  40  42  44 
Wattless  Component  of  Current 

FIG.  94. — Diagram  for  Obtaining  Rotor  Current  at  Half-load. 

The  readiest  way  to  obtain  the  corresponding  value  of  the 
secondary  current  is  to  make  the  circle-diagram  construction 
indicated  in  Fig.  94.  From  this  construction  we  see  that  for  a 
1  :  1  ratio,  the  secondary  current  would  be  47.5  amperes.  We 


180       POLYPHASE  GENERATORS  AND  MOTORS 


have  seen  on  page  165  that  at  full  load,  when  the  1  :  1  secondary 
current  was  96  amperes,  the  rotor  PR  loss  was  3140  watts. 
Consequently  at  half  load,  the  rotor  PR  loss  will  be: 


X  3140  =  770  watts. 


Thus  at  half  load  we  have : 

Stator  PR  loss ' 1  340  watts 

Stator  core  loss 2  410     " 

Rotor  PR  loss 770     " 

Rotor  core  loss 240     ' ' 

Friction  and  windage  loss 1  300     ' ' 


Total  of  all  losses 6  060  watts 

Output  at  half  load. 74  600     " 


Input  at  half  load .   80  660  watts 

74-  fiOO 

Half-load  efficiency  =»  ^    ^  X 100  =  92.5  per  cent. 
oU  DOu 

Making  similar  calculations  for  other  loads  we  obtain  the  follow 
ing  inclusive  table  of  the  losses  at  various  loads : 

TABULATION  OF  LOSSES  AND  EFFICIENCIES  AT  60°  CENT. 


Percentage  of  rated  out- 
put 

0 

25 

50 

75 

100 

125 

Amperes  input  per  phase 

25.2 

35.0 

55.5 

77.0 

102 

127 

Stator  I2R  loss           .... 

280 

550 

1340 

2580 

4540 

7100 

Stator  core  loss  
Rotor  72B  loss  
Rotor  core  loss. 

2410 
20 
240 

2410 
300 
240 

2410 
770 
240 

2410 
1750 
240 

2410 
3140 
240 

2410 
4900 
240 

Friction  and  windage  loss 

1300 

1300 

1300 

1300 

1300 

1300 

Total  of  all  losses  
Output  in  watts  

4250 
0 

4800 
37400 

6060 
74600 

8280 
112000 

11630 
149200 

15950 
186600 

Output  in  horse-power  .  . 

0 

50 

100 

150 

200 

250 

Input 

4250 

42200 

80660 

120280 

160830 

202550 

0 

0.885 

0.925 

0  930 

0  930 

0  921 

These  efficiencies  are  plotted  in  the  curve  of  Fig.  95. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   181 

Estimation  of  the  Temperature  Rise.  In  making  a  rough 
estimate  of  the  heating  of  the  motor  the  first  step  consists  in  cal- 
culating the  "  watts  per  square  decimeter  (sq.dm.)  of  radiating 
surface  at  the  air-gap." 

The  "  radiating  surface  at  the  air-gap  "  is  taken  as  the  sur- 
face over  the  ends  of  the  stator  windings.  While  this  surface 
will  vary  considerably  according  to  the  type  of  winding  employed, 
a  representative  basis  may  be  obtained  from  the  formula: 


"  Equiv.  radia.  surface  at  air-gap  " 


100 

1    90 

2 
® 
P4  80 

d 

I  70 

.2 

I  G° 
50 


20       40       60      80      100     120    140     160     180     200     220     240    260    280 
Output  in  Horse-power 

FIG.  95. — Efficiency  Curve  for  200-H.P.  Polyphase  Induction  Motor. 


The  k  in  this  formula  is  a  factor  which  is  a  function  both  of 
the  normal  pressure  for  which  the  motor  is  built,  and  of  the  polar 
pitch,  T.  Suitable  values  will  be  found  in  Table  32. 

TABLE  32. — DATA  FOR  ESTIMATING  THE  RADIATING  SURFACE. 


Rated  Pressure  of  the  Motor. 

Values  of  k  in  the  Formula: 
Equiv.  rad.  sur.  =irXDX(\g  +kr). 

T=60  cm. 

r  =40  cm. 

r=20  cm. 

1000  volts  (or  less).  .  .  
2000           

0.8 

0.9 
1.0 

1.1 

1.2 
1.4 

1.7 

0.9 
.0 
.1 

.3 
.5 

.8 

2.0 

1.1 
1.2 
1.4 

1.7 
2.0 
2.3 

2.8 

4000           

6000           

8000 

10000           

12000     '  '     

182       POLYPHASE  GENERATORS  AND  MOTORS 
For  our  design  we  have: 

D  =  62.0; 
X0-43.0; 

«r  =  32.5; 


.'.     "  Equivalent  radiating  surface  at  air-gap 

=  xX62.0X  (43.0+29.3) 
=  xX62.0X72.3 
=  14000  sq.cm. 
=  140  sq.dm. 

The  loss  to  be  considered  is,  at  full  load: 

11  630  watts. 
Thus  we  have: 

Watts  per  sq.dm.  =  —  T-  =83.4. 


The  data  in  Table  33,  gives  a  rough  notion  of  the  temperature 
rise  corresponding  to  various  conditions. 

TABLE  33.  —  DATA  FOR  ESTIMATING  THE  TEMPERATURE  RISE  IN  INDUCTION 

MOTORS. 


Peripheral 
Speed  in 
Meters  per 
Second. 

Thermometrically  determined  ultimate  temperature  rise  of  open-protected 
types  of  induction  motors,  per  watt  of  total  loss  per  sq.dm.  of  equiva- 
lent radiating  surface  at  the  air-gap. 

Badly  Ventilated. 

Fairly  Ventilated. 

Very  Well  Ventilated. 

10 
20 
30 

40 
50 
60 

0.59 
0.49 
0.47 

0.45 
0.43 
0.41 

0.41 
0.40 
0.38 

0.36 
0.34 
0.32 

0.36 
0.34 
0.32 

0.29 
0.26 
0.24 

Our  motor  is  decidedly  well  supplied  with  ventilating  ducts, 
and  with  reasonable  care  in  the  arrangement  of  parts,  it  will 
come  in  the  class  of  "  very  well  ventilated  "  motors.  The  per- 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  183 

ipheral  speed  has  already  (on  page  113)  been  ascertained  to  be 
16.2  meters  per  second. 

From  Table  33  we  find  that  the  value  for  the  thermometric- 
ally  determined  temperature  rise  is: 

0.35°  per  watt  per  sq.dm. 

This   gives   us   a   total    (thermometrically   determined)    rise   of 
83.4X0.35  =  29°  Cent. 

Although  in  the  predetermination  of  temperature  rises,  no 
close  accuracy  is  practicable,  still,  the  margin  indicated  by  the 
above  result  is  so  great  that  the  designer  should  rearrange  the 
final  design  in  such  a  manner  as  to  save  a  little  material  at  the 
cost  of  a  slight  increase  in  the  losses. 

Since  a  rise  of  fully  40°  cent,  is  usually  considered  quite  con- 
servative (45°  often  being  adopted),  an  estimated  rise  of  35° 
would  leave  a  sufficient  margin  of  safety. 

THE  WATTS  PER  TON 

Another  useful  criterion  to  apply  in  judging  whether  a  motor 
is  rated  at  as  high  an  output  as  could  reasonably  be  expected,  is 
the  "  watts  per  ton."  The  weight  taken,  is  exclusive  of  slide  rails 
and  pulley.  In  the  case  of  a  design  which  has  not  been  built, 
the  method  of  procedure  is  as  follows : 

First  estimate  the  weights  of  effective  material. 

Weight  Stator  Copper. 

mlt.  =  183.5  cm. 

Total  number  of  turns  =  3  T  =  3  X 120  =  360; 

Cross-section  stator  cond    =0.302  sq.cm.; 
Volume  of  stator  copper      =  183.5X360X0.302 

=  19900  cu.cm.; 
Weight  1  cu.cm.  of  copper  =8.9  grams; 

w  .  ,  .  19900X8.9 

Weight  stator  copper  = 

J-UUu 

=  177  kg. 


184       POLYPHASE  GENERATORS  AND  MOTORS 

Weight  Rotor  Copper. 

Total  length  rotor  face  1  _QOon 

1       /  -n-rr\          '    —  O^yU   CHI.; 

conds  (see  page  175)     J 

Cross-section  =5.4X0.48  =  2.59  sq.cm.; 

Volume  of  rotor  face  conds  =  3290X2.59 

=  8550  cu.cm.; 

8550X8.9 
Weight  rotor  face  conds      = 


=  76  kg.; 

Mean  circum.  end  ring        =177  cm.; 
Cross-section  =15.1  sq.cm.; 

2X177X15.1X8.9 
Weight  two  end  rings          =  — 

1UUU 

=  47.5  kg.; 

Total  weight  rotor  copper  =76+48  =  124  kg.; 
Total     weight      copper    I_177,l9d 

,     .  i  x  f   —  -L  I  I     |~  JL^iTt 

(stator  plus  rotor)   j 

=  301  kg. 

Weight  Stator  Core. 

This  has  already  been  estimated  (on  page  159)  to  be  603  kg. 

Weight  Rotor  Core. 

External  diameter  rotor  core  =61.7  cm.; 

Internal  diameter  rotor  core  =38.3  cm.; 

Gross  area  rotor  core  plate  =  |(61.72-38.32) 

=  1840  sq.cm.; 

Area  70  slots          =  70  X  5.4  X  0.48  =   180  sq.cm.  ; 
Net  area  rotor  core  plates  =  1660  sq.cm.; 

Aw  =  27.9  cm.; 
Volume  of  sheet  steel  in  rotor  core  =  1660X27.9 

=  46  300  cu.cm.; 
Weight  of  1  cu.cm.  of  sheet  steel   =7.8  grams; 

46300X7.8 
Weight  of  rotor  core 


=  360  kg.; 
Total  weight  sheet  steel  (stator 


,  603 +360  =  963  kg. 

plus  rotor) 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  185 


Thus  we  have: 

Weight  copper 
Weight  sheet  steel 


=  301  kg.; 
=  963kg. 


Total  net  weight  effec.  material      =  1264  kg. 

From  experience  in  the  design  of  motors  of  this  type  it  has  been 
determined  that  an  adequate  mechanical  design  is  consistent  with 
obtaining  for  the  ratios  of  "  Total  Weight  of  Motor  "  to  "  Weight 
of  Effective  Material,"  the  values  given  in  Table  34. 

TABLE  34. — DATA  FOR  ESTIMATING   THE  TOTAL  WEIGHT  OF  AN  INDUCTION 

MOTOR. 


D   • 
Diameter 
at  Air-gap 
(in  cm.). 

Ratio  of  Total  Weight  (exclusive  of  slide 
rails  and  pulley)  to  Weight  of  Net 
Effective  Material. 

D/X,  =  1.5. 

Z)/X,  =  2.5. 

20 

1.80 

2.20 

40 

1.60 

1.82 

80 

1.54 

1.78 

For  our  motor  we  have: 


Total  weight   (exclusive  of  slide  rails  and  pulley)  =  1.56X1264 
-1970kg. 

=  1.97  tons. 

It  will  be  remembered  (p.  Ill)  that  our  original  rough  estimate 
for  the  weight,  was  2.0  tons.  Even  this  last  method,  which 
depends  upon  obtaining  a  factor  by  which  to  multiply  the  cal- 
culated weight  of  effective  material,  can  only  be  regarded  as  rough. 
We  can,  however,  regard  the  value  of  1.97  tons  as  fairly  correct 
for  the  total  weight  of  our  motor. 


186       POLYPHASE  GENERATORS  AND  MOTORS 
Consequently  we  have : 

Watts  per  ton  =ry|^  =  5900. 

The  designer  must  gain  his  own  experience  as  to  the  attainable 
values  for  the  "  watts  per  ton."  It  may,  however,  here  be  stated 
that  8000  watts  per  ton  and  even  considerably  higher  values  are 


FIG  96. — An  8-pole,  50-H.P.,  900-r.p.m.,  Squirrel-cage  Induction  Motor  of 
the  Open-protected  Type.     [Built  by  the  General  Electric  Co.  of  America.! 

quite  consistent  with  moderate  temperature  rise  for  a  motor  of 
this  size  and  speed,  if  constructed  in  the  manner  generally 
designated  the  "  open-protected  type."  In  Fig.  96  is  given  a 
photograph  of  an  open-protected  type  of  squirrel-cage  induction 
motor. 

The  Breakdown  Factor.  It  has  been  stated  that  so  far  as 
temperature-rise  is  concerned,  the  motor  is  rather  liberally 
designed  and  that  in  a  revised  design  it  should  be  possible  to  save 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   187 

a  little  in  material.  We  must,  however,  take  into  account  the 
limitation  imposed  by  the  heaviest  load  which  the  motor  is 
capable  of  carrying  before  it  pulls  up  and  comes  to  rest.  The 
ratio  of  this  load  to  the  rated  load  is  termed  the  breakdown 
factor  (bdf.)  and  the  present  custom  is  to  require  that  a  motor 
shall  have  a  breakdown  factor  of  about  2.  In  the  case  of  our 
200-h.p.  motor,  it  ought  to  be  capable  of  carrying  an  instantaneous 
load  of  2X200  =  400  h.p.  without  pulling  up  and  coming  to  rest. 
It  has  previously  been  explained  that  the  vertical  projection  of 
the  vector  representing  the  stator  current  in  the  circle  diagram, 
is,  for  constant  terminal  pressure,  proportional  to  the  watts  input 
to  the  motor.  Thus  the  current  corresponding  to  the  maximum 
input  is  the  current  AM  in  the  diagram  in  Fig.  97.  The  power 
input  corresponding  to  this  current  may  be  represented  by  MN, 
the  vertical  radius  of  the  circle.  We  have: 

=  VAN2+MN2 


=  V3332+3082 
=  V206000 
=  454  amperes. 

Since  the  phase  pressure  is  577  volts,  we  have: 

Power  input  =  3  X  577  X  308 
=  534000  watts. 

Consequently  when  used  as  a  measure  of  the    power  input, 
the  vertical  ordinates  are  to  the  scale  of: 

3X577  =  1730  watts  per  ampere. 

But  it  is  not  the  input  which  we  wish,  but  the  output.     The 
PR  losses  at  the  full-load  input  of  102  amperes,  amount  to: 

4540+3140  =  7680  watts. 


188       POLYPHASE  GENERATORS  AND  MOTORS 


Consequently  when  the  input  is  454  amperes,  the  I2R  losses 


are: 


~     X  7680  =  152  000  watts. 


The  core  losses  and  friction  aggregate: 

2410+240+1300  =  3950  watts. 


240 
220 


.180 
§160 
ft140 


§100 
W  80 
60 
40 
20 
A 


20    40    60   80  100  120  140  160  180  200  220  240  260  280  300  320  340  360  380  400 
Wattless  Component  of  Current 


FIG.  97. — Diagram  for  Preliminary  Consideration  of  the  Breakdown  Factor. 

The  total  internal  losses  amount  to: 

152  000+4000  =  156  000  watts. 

Thus  the  output  corresponds  to  that  portion  of  the  vertical 
radius  MN,  which  is  equal  to : 

534  000  - 156  000  =  378  000  watts. 
This  is  the  vertical  height,  PN,  laid  off  equal  to : 

378  000 
--  =  218  amperes. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  189 

This  corresponds  to  an  output  of: 

378  000 


746 


=  505h.p. 


This  vertical  radius  does  not,  however,  correspond  to  the  point 
of  maximum  output.  Let  us  draw,  as  in  Fig.  98,  the  vectors 
corresponding  to: 

•  430,  400     and    370  amperes. 


.320 
300 


^240 
2220 
|200 
"olSO 
§160 

0,140 
g 

°120 


60 


20 


—I 


20    40   60    SO  100  120  140  160  ISO  200  220  240  260  280  300  -SO  340  360  380  400-420440 
Wattless  Component   of  Current 

FIG.  98. — Revised  Diagram  for  Determining  the  Breakdown  Factor. 


The  corresponding  vertical  ordinates  are: 

307,  300    and    290  amperes. 
The  inputs  are: 

1730X307  =  531  000  watts, 
1730X300  =  520  000  watts, 
1730X290  =  502  000  watts. 


190       POLYPHASE  GENERATORS  AND  MOTORS 
The  /2#  losses  are: 


2 
X7680  =  137  00°  watts> 


(4on\2 
Y~j  X  7680  =  118  000  watts, 


=  101  000  watts. 


Adding  the  remaining  losses  of  4000  watts,  we  obtain  for  the 
total  losses  in  the  three  cases: 

141  000,  122  000,  and  105  000  watts. 

Deducting  these  losses  from  the  respective  inputs,  we  obtain  as 
the  three  values  for  the  output: 

531  000-141  000  =  390  000  watts, 
520  000-122  000  =  398  000  watts, 
502  000  -  105  000  =  397  000  watts. 

Obviously  the  output  is  a  maximum  at  an  input  of  400  amperes 
and    then    amounts    to: 

398  000 


Thus  for  our  motor  the  bd.f.  is: 

S 

A   rough,    empirical   formula   for   obtaining   the   breakdown 
factor  is: 

•*  -f. 

For  our  motor  we  have  : 

y  =  0.248; 


.'.   bdf= 


0.041. 
0.4X0.24? 


0.041 
=  2.42. 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR  191 

The  use  of  this  empirical  formula  is,  in  general,  quite  sufficient, 
as  it  is  rarely  of  consequence  to  be  able  to  estimate  the  bdf.  at 
all  closely. 


THE  POWER-FACTOR 

It  is  a  very  simple  matter  to  obtain  a  curve  of  power-factors 
for  various  values  of  the  current  input.     In  Fig.   99  are  drawn 


320 


240 

220 

200 

180 

160 

140 

120 

100 

80 

60 

40 

20 


Wattless  Component  of  the  Current 

FIG.  99. — Diagram  for  Determining  the  Power-factors. 


vectors  representing  stator  currents  ranging  from  the  no-load 
current  of  25.2  amperes,  up  to  the  break-down  current  of  400 
amperes.  In  each  case,  the  power-factor  is  the  ratio  of  the 
vertical  projection  of  the  stator  current,  to  the  stator  current. 
The  values  of  the  currents  and  of  the  vertical  projections  are 
recorded  in  Fig.  99,  and  also  in  the  first  two  columns  of  the 
following  table  in  which  the  estimation  of  the  power-factor  is 
carried  out,  and  also  the  estimation  of  the  efficiencies. 


192       POLYPHASE  GENERATORS  AND  MOTORS 


I. 

II. 

III. 

IV. 

V. 

VI. 

VII. 

VIII. 

Stator 
Current 

Phlse. 

Vertical 
Com- 
ponent. 

Power 
Factor, 
(II  -i-1). 

Input  in  Watts 
(3X577  XII). 

Losses. 

Output 
in  Watts, 
(IV  -V). 

Output 
in  H.P. 

Efficiency, 
/VIX100\ 

(rv      )• 

25.2 

2.3 

0.09 

4000 

4000 

0 

0 

0 

70 

61 

0.87 

105  000 

7600 

97400 

130 

0.93 

120 

110 

0.92 

190000 

14600 

175400 

235 

0.92 

170 

155 

0.91 

268000 

25300 

243000 

326 

0.91 

220 

197 

0.90 

342000 

39700 

302000 

405 

0.88 

270 

235 

0.87 

406000 

57800 

348  000 

465 

0.86 

320 

264 

0.83 

457  000 

79300 

378  000 

506 

0.83 

370 

290 

0.78 

502  000 

105000 

397  000 

531 

0.79 

400 

300 

0.75 

520000 

122000 

398000 

534 

0.77 

In  Figs.  100  and  101,  these  values  for  the  power-factor  and 
efficiency  are  plotted  as  ordinates  and  with  amperes  per  phase 
as  abscissae.  In  Figs.  102  and  103  they  are  again  plotted  as 
ordinates  but  with  the  output  in  horse-power  as  abscissae. 


i.UU 

0.90 
0.80 
0.70 
|0.60 
^0.50 
§0.40 
0.30 
0.20 
0.10 

2 

~~—  i~. 

^ 

^ 

\ 

50 


100   150    200    250    300    350  400 
Amperes  Input 


FIG.    100. — Efficiency    Curve    with 
Current  Input  as  Abscissae. 


FIG.  101. — Power-factor  Curve  with 
Current  Input  as  Abscissae. 


THE  SLIP 

Were  it  not  that  the  apparent  resistance  of  the  slot-embedded 
portions  of  the  conductors  of  the  squirrel  cage,  varies  with  the 
periodicity  of  the  rotor  currents,  the  determination  of  the  slip  at 
any  load  would  be  a  very  simple  matter.  The  effect  of  this 


POLYPHASE  MOTOR  WITH  SQUIRREL-CAGE  ROTOR   193 


variation  is  too  small  to  be  of  consequences  at  ordinary  loads  and 
may  for  most  purposes  be  neglected.  For  constant  apparent 
resistance,  percentage  slip  is  the  percentage  which  the  rotor 
PR  loss  constitutes  of  the  input  to  the  rotor. 

The  Determination  of  the  Rotor  I2R  Loss  at  Various  Loads. 
By  constructions  similar  to  that  illustrated  in  Fig.  94  on  p.  179, 
we  may  obtain  the  values  of  the  rotor  current  for  various  values 
of  the  stator  current.  A  number  of  such  values  is  tabulated 
below: 


Stator  Current. 

Rotor  Current  for  a 
1:1  Ratio. 

Actual  Rotor 
Current  (Ratio  of 
Trans,  is  10.3  :  1). 

25.2  amp. 
70    " 

0  amp. 
62    " 

0  amp. 

640    " 

120    " 

113    " 

1160    " 

170    " 

162     M 

1670    " 

220    " 

210    " 

2160    " 

270    " 

256     " 

2640    " 

320    " 

305    " 

3140    " 

370    " 

354    " 

3640    " 

400    " 

384    " 

3960    " 

100 


Output  in  Horsepower 

FIG.    102. — Efficiency    Curve    with 
Horse-power  Output  as  Abscissae. 


!    r-i    w    cq    eo    eo     -^    TJI    10 

Output  in  Horsepower 

FIG.  103. — Power-factor  Curve  with 
Horse-power  Output  as  Abscissae. 


With  the  full-load  stator  current  of  102  amperes  we  have  seen 
(on  p  166.)  that  the  squirrel-cage  loss  is  3140  watts.  From 
the  above  table  we  see  that  for  a  stator  current  of  102  amperes, 

the  rotor  current  is  l^:XHQQ  =  j  985  amperes.     Thus,  referred 


194       POLYPHASE  GENERATORS  AND  MOTORS 


to  the  current  in  the  face  conductors,  the  squirrel  cage  may  be 
considered  as  having  an  "  equivalent  "  resistance  of: 
3140 


9852 


=  0.00324  ohm. 


In  the  following  table,  the  rotor  I2R  losses,  the  input  to  the 
rotor,  the  slip  and  the  speed  have  been  worked  out: 


6 

Q»  £ 

I 

u  . 

| 

0-2  .? 

S  of 

05   <-< 

'~  ^  03  0, 

e 

S  x 

liT 

|* 

"o  o 

S3 

ts  £ 

02 

o  §6 
1  S3 

|| 

02 

a 

flls 

gOOO 

o   w 

|l 

tf  S 

IH  I-  . 

0 

V 

[Al 

[B] 

[C] 

[D] 

[E] 

[F] 

to; 

500 

25.2 

280 

2410 

4000 

1310 

0 

0 

0 

0 

70 

2140 

2410 

105  000 

100  450 

640 

1330 

1.32 

130 

493 

120 

6300 

2410 

190  000 

181  300 

1160 

4350 

2.40 

235 

488 

170 

12600 

2410 

268  000 

253  000 

1670 

9000 

3.55 

326 

482 

220 

21  100 

2410 

342  000 

318  500 

2160 

15100 

4.75 

405 

476 

270 

32000 

2410 

406  000 

371  600 

2640 

22500 

6.05 

465 

469 

320 

44800 

2410 

457000 

409  800 

3140 

32000 

7.81 

506 

461 

370 

60000 

2410 

502  000 

439  600 

3640 

43000 

9.80 

531 

451 

400 

70000 

2410 

520  000 

447  600 

3960 

51  000 

11.4 

534 

443 

M 

22 
20 
18 
16 
1  14 
I12 

I  10 

*     8 
6 
4 
2 

1 

I 

/ 
/ 

} 

1 

/ 

I 

/ 

1 

/ 

/ 

/ 

xx 

x^^* 

Y^ 

^-^ 

^ 

-^^ 

,^-~ 

—  — 

DUU 

500 
400 
300 
200 
100 



••-JT- 

-  = 

~ 

r"V 

—  ^^, 

' 

\ 

\ 

100       200       300      400       500 
Output  in  Horsepower 

FIG.  104.— Curves  of  Slip. 


100       200       300        400       500 
Output  in  Horsepower 

FIG.  105. — Curves  of  Speed. 


These  values  for  "  slip  "  and  speed  are  plotted  in  the  full- 
line  curves  in  Figs.  104  and  105;  the  dotted  line  curves  being  added 
to  indicate  qualitatively  the  general  way  in  which  the  results 
would  be  affected  by  the  influence  of  the  frequency  of  the  rotor 
currents  in  increasing  the  apparent  resistance  of  the  slot-embedded 
portions  of  the  squirrel-cage. 


CHAPTER  V 
SLIP-RING  INDUCTION  MOTORS 

PROBABLY  considerably  over  90  per  cent  of  the  total  number 
of  induction  motors  manufactured  per  annum  are  nowadays 
of  the  squirrel-cage  type.  The  strong  preference  for  this  type 
is  on  account  of  the  exceeding  simplicity  of  its  construction  and 
the  absence  of  moving  contacts.  Nevertheless  occasions  arise 
when  it  is  necessary  to  supply  definite  windings  on  the  rotor 
and  to  connect  these  up  to  slip  rings.  Sometimes  this  is  done 
in  order  to  control  the  starting  torque  and  to  obtain  a  higher 
starting  torque  for  a  given  stator  current  than  could  be  obtained 
with  a  squirrel-cage  motor  and  sometimes  the  use  of  slip  rings 
is  for  the  purpose  of  providing  speed  control  by  regulating  external 
resistances  connected  in  series  with  the  rotor  windings.  This 
is  a  very  inefficient  method  of  providing  speed  control,  but  cases 
occasionally  arise  when  it  is  the  economically  correct  method  to 
employ. 

Since  the  controlling  resistance  is  located  external  to  the  motor, 
there  is  no  longer  occasion  to  study  to  provide  a  deep  rotor  con- 
ductor in  order  to  obtain  a  desired  amount  of  starting  torque. 
With  freedom  from  this  restriction,  the  density  at  the  root  of  the 
rotor  tooth  would  (for  a  design  of  the  rating  we  have  discussed 
in  Chapter  IV)  be  taken  higher  than  the  15  500  lines  per  sq.  cm. 
adopted  for  the  squirrel-cage  design.  We  shall  do  well  to  employ 
a  somewhat  shallower  and  wider  slot,  for  it  must  be  observed  that 
it  now  becomes  necessary  to  provide  space  for  insulating  the  rotor 
conductors. 

Let  us  employ  70  slots,  as  before,  but  let  the  slots  have  a  depth 
of  only  30  mm.  and  a  (punched)  width  of  8  mm.  The  diameter 
at  the  bottom  of  the  slots  will  now  be: 

617.4-2X30  =  557.4  mm. 

195 


196       POLYPHASE  GENERATORS  AND  MOTORS 
The  rotor  slot  pitch  is  now: 

557.4  XTC     OK  n 
-njQ — =25.0  mm. 

Width  tooth  at  root  =  25.0 -8.0  =  17.0  mm. 

For  the  squirrel-cage  design,  the  slot  pitch  (see  p.  888)  was 
22.8  mm.  and  the  width  of  tooth  at  the  root,  was: 

22.8 -4.8  =  18.0  mm. 

Thus  the  tooth  density  will  be  increased  in  the  ratio  of  17  to 
18,  but  since  the  length  of  the  tooth  has  been  decreased  in  the 
ratio  of  54  to  30,  the  mmf .  required  for  the  rotor  teeth  will  be  no 
greater  than  for  the  squirrel-cage  design. 

The  thickness  of  the  slot  insulation  will  be  0.8  mm. 

Allowing  0.3  mm.  tolerance  in  assembling  the  punchings,  the 
assembled  width  of  the  slot  will  be: 

8.0-0.3  =  7.7  mm. 

After  deducting  the  portions  of  the  width  occupied  by  the  slot 
insulation  on  each  side,  we  arrive  at  the  value  of : 

7.7-2X0.8  =  6.1  mm. 

for  the  width  of  the  conductors,  of  which  there  will  be  2  per  slot, 
arranged  one  above  the  other.  Of  the  depth  of  30  mm.,  the 
insulation  and  the  wedge  at  the  top  of  the  slot  will  require  a  total 
allowance  of  6  mm.  Consequently  for  the  depth  of  each  conduc- 
tor we  have: 

30-6    \  , 

=  )  12  mm. 

Thus  each  conductor  has  a  height  of  12  mm.,  a  width  of  6.1 
mm.  and  a  cross-section  of : 

12X6.1=73  sq.mm. 


SLIP-RING  INDUCTION  MOTORS 


197 


A  drawing  of  the  slot  is  given  in  Fig.  106. 

A  two-layer,  full-pitch,  lap  winding  will  be  employed.  There 
are  70  slots  and  (2X70  =  )  140  conductors.  The  winding  will 
be  of  the  2-circuit  type.  It  is  impracti- 
cable in  this  treatise  to  deal  with  the 
extensive  subject  of  armature  windings. 
The  reader  will  find  a  discussion  of  the 
laws  of  2-circuit  windings  on  page  3  of 
the  Author's  "  Elementary  Principles  of 
Continuous-Current  Dynamo  Design  " 
(Whittaker  <fc  Co.,  London  and  New 
York),  and  also  in  Chapter  VIII  of  Hobart 
and  Ellis'  "  Armature  Construction  " 
(Whittaker  &  Co.).  It  must  suffice  here 
to  state  that  such  a  winding  is  usually 
arranged  to  comply  with  the  formula: 

F  =  Py±2. 

In  this  formula,  the  total  number  of  conductors  is  denoted  by 
F,  the  number  of  poles  by  P,  and  the  "  winding  pitch  "  by  y. 
The  conductors  are  numbered  from  1  to  140  as  indicated  in  the 
diagram  in  Fig.  107.  The  winding  pitch,  yt  is  that  quantity 
which  is  added  to  the  number  of  any  conductor  in  order  to  ascer- 
tain the  number  of  the  conductor  to  which  the  first  conductor 
is  connected. 

For  our  case  we  have: 


FIG. 
"We 

z: 

"s: 

i 

1 

-; 

;}.-!* 

•  7i7  

106.—  S 
>und"   I 

lot  of 
totor. 

Consequently  conductor  number  1  is  connected  (as  indicated 
in  Fig.  107),  to  conductor  number  (1+23  =  )  24;  conductor  number 
24  to  conductor  number  (24+23  =  )  47,  etc.,  until  the  entire 
140  conductors  are  interconnected  to  constitute  a  singly-re- 
entrant  circuit.  After  the  diagram  has  been  carried  out  in 
this  way  in  pencil,  the  next  step  consists  in  dividing  up  the 

(-—=)    70  turns  into  six  separate  circuits.     The  six  circuits 


198       POLYPHASE  GENERATORS  AND  MOTORS 

cannot  contain  equal  numbers  of  turns.  The  nearest  to  this 
consists  in  subdividing  the  winding  up  into  two  circuits  each 
comprising  22  conductors  and  four  circuits  each  comprising  24 
conductors.  We  have 

22+22+24+24+24+24  =  140. 


FIG.  107. — Winding  Diagram  for  the  Rotor  of  a  Slip-ring  Induction  Motor. 


Let  us  start  in  with  conductor  number  1  and  designate  this 
beginning  by  A\,  as  shown  in  Fig.  107.  After  following  through 
the  22  conductors: 

(1,  24,  47,  70,  93,  116,  139,  22,  45,  68,  91,  114,  137, 
20,  43,  66,  89,  112,  135,  18,  41,  64), 


SLIP-RING  INDUCTION  MOTORS  199 

we  interrupt  the  winding  and  bring  out  a  lead  at  the  point  Am. 
We  then  start  in  again,  indicating  the  point  as  B\  and  proceed 
next  through  conductor  number  (64+23  =  )  87.  After  passing 
through  22  more  conductors  we  come  out  again  at  a  point  which 
we  designate  as  Bm.  'the  next  circuit  starts  at  C\  and  ends  at 
Cm.  The  remaining  three  circuits  are  A^An,  B^Bn  and  C<iCn- 

We  now  have  six  circuits.  An  inspection  will  show  that  these 
can  be  grouped  in  three  pairs  which  have,  in  Fig.  107  been 
drawn  in  red,  black  and  green.  The  red  phase  comprises  A\Am 
connected  in  series  with  AnA%  forming  A\A^.  In  the  same  way, 
the  black  phase  B\B^  is  obtained,  and  the  green  phase  CiCz.  By 
considering  the  instant  when  the  current  in  Phase  A  is  of  the  value 
1.00  and  is  flowing  from  the  line  toward  the  common  connection, 
while  the  currents  in  Phases  B  and  C  are  of  the  value  0.50  and 
are  flowing  from  the  common  connection  to  the  lines,  it  is  ascer- 
taineol  that  the  ends  A%,  B\  and  Cz  must  be  brought  together  to 
form  the  common  connection,  the  ends  A\,B%  and  Ci  constituting 
the  terminals  of  the  motor. 

For  the  mean  length  of  turn  of  this  rotor  winding  we  have  the 
formula  : 

mlt.~2XH-2.5Tj 


T  =  32.5; 

2.5T  =  82; 

mlt.  =  86+82  =  168  cm. 

The  entire  winding  comprises  (70X2  =  )  140  conductors  and 
70  turns.     Consequently  the  average  length  per  phase  is: 

70X168 
—^  -  =3920  cm. 

o 

x  N     3920X0.00000200 
Resis.  per  phase  (at  60°  Cent.)  =  -       —zr^  —  —  =  0.0107  ohm. 

U.  /o 

Since  there  are  720  stator  conductors,  the  ratio  of  transforma- 
tion is  now  : 


^-1  5.  15  to  1.00. 


200       POLYPHASE  GENERATORS  AND  MOTORS 

Referring  to  p.  165,  we  see  that  for  a  1  :  1  ratio,  the  rotor 
current  corresponding  to  the  full-load  stator  current  of  102 
amperes,  is  96  amperes.  Consequently  the  full-load  rotor  current 
for  our  wound  rotor,  is: 

5.15X96  =  495  amperes. 
The  full  load  PR  loss  in  the  rotor  is: 

3X4952X0.0107  =  7800  watts. 

/7800     \ 
This  is  (5777:=  )2.5  times  the  loss  in  our  squirrel-cage  design. 

\O-L4:U         J 


The  increase  is  due  partly  to  the  waste  of  space  for  slot  insulation 
and  partly  to  the  long  end  connections  inevitably  associated  with 
a  "  wound  "  rotor. 

These  long  end  connections  also  involve  additional  magnetic 
leakage  at  load,  and  the  circle  ratio  of  a  motor  with  a  "  wound" 
rotor  may  usually  be  estimated  on  the  basis  of  a  25  per  cent 
increase  in  the  values  in  Table  27,  on  pp.  150  and  151.  Conse- 
quently for  the  present  design  we  have  : 

<j=  1.25X0.041  =0.051. 

We  still,  have  (as  on  p.  140),  y  =  0.242. 
Hence  for  the  circle  diameter  we  now  have: 


TV       t    -    i       0.242X102 
Dia.  of  circle  =  — ^r^\ —  =  485  amperes 
u.uoi 


The  "  ideal  "  short-circuit  current  is  now: 

0.242X102+485  =  510  amperes. 

as  against  the  value  of  645  amperes  for  the  squirrel-cage  design. 
We  now  have : 


SLIP-RING  INDUCTION  MOTORS  201 

From  an  examination  of  these  various  ways  in  which  the 
properties  of  the  motor  have  suffered  by  the  substitution  of  a 
wound  rotor,  it  will  be  understood  that  the  squirrel-cage  type 
is  much  to  be  preferred,  and  that  every  effort  should  be  made  to 
employ  it  when  the  conditions  permit.  In  addition  to  the 
defects  noted,  it  must  not  be  overlooked  that  the  efficiency  and 
power-factor  of  the  slip-ring  motor  are  lower  and  the  heating 
greater.  Furthermore  the  squirrel-cage  motor  is  more  compact 
and  is  lighter  and  cheaper.  It  will  also  be  subject  to  decidedly 
less  depreciation,  and  can  be  operated  under  conditions  of 
exposure  which  would  be  too  severe  for  a  slip-ring  motor. 


CHAPTER  VI 
SYNCHRONOUS  MOTORS  VERSUS  INDUCTION  MOTORS 

THE  circumstance  that  a  type  of  apparatus,  possesses  particu- 
larly attractive  features,  is  liable  to  lead  occasionally  to 
disappointment,  owing  to  its  use  under  conditions  for  which  it  is 
rendered  inappropriate  by  its  possession  of  other  less-well- 
known  features,  which,  under  the  conditions  in  question,  are 
undesirable. 

While  the  squirrel-cage  induction  motor  is  of  almost  ideal 
simplicity,  there  are  many  instances  in  which  its  employment 
would  involve  paying  dearly  for  this  attribute.  It  is,  for  instance, 
a  very  poor  and  expensive  motor  for  low  speeds.  When  the 
rated  speed  is  low,  and  particularly  when,  at  the  same  time,  the 
periodicity  is  relatively  high,  the  squirrel-cage  motor  will 
inevitably  have  a  low  power-factor.  In  Figs.  108  and  109  are 
drawn  curves  which  give  a  rough  indication  of  the  way  in  which 
the  power-factor  of  the  polyphase  induction  motor  varies  with 
the  speed  for  which  the  motor  is  designed.  While  qualitatively 
identical  conclusions  will  be  reached  by  an  examination  of  the 
data  of  the  product  of  any  large  manufacturer,  the  quantitative 
values  may  be  materially  different,  since  each  manufacturer's 
product  is  characterized  by  variations  in  the  degree  to  which 
good  properties  in  various  respects  are  sacrificed  in  the  effort 
to  arrive  at  the  best  all-around  result.  It  is  for  this  reason  that 
instead  of  employing  data  of  the  designs  of  any  particular  manu- 
facturer, the  curves  in  Figs.  108  and  1C 9  have  been  deduced  from 
the  results  of  an  investigation  published  by  Dr.  Ing.  Rudolf 
Goldschmidt,  of  the  Darmstadt  Technische  Hochschule,  in  an 
excellent  little  volume  which  he  has  published  under  the  title: 
"  Die  normalen  Eigenschaften  elektrischer  Maschinen  "  (Julius 
Springer,  Berlin).  Whereas  from  the  50-cycle  curves  of  Fig.  109, 
it  will  be  seen  that,  at  low  speeds,  motors  of  from  50  to  500  h.p. 

202 


SYNCHRONOUS  MOTORS  vs.  INDUCTION  MOTORS    203 

rated  output,  have  power-factors  of  less  than  0.80,  the  curves 
in  Fig.  108  show  that  the  power-factors  of  equivalent  25-cycle 
motors  are  distinctly  higher. 

It  is  desirable  again  to  emphasize  that  the  precise  values 
employed  in  the  curves  in  Figs.  108  and  109,  have,  considered 
individually,  no  binding  significance.  Thus,  by  sacrificing 
desirable  features  in  other  directions  and  by  increased  outlay 
in  the  construction  of  the  motor,  slightly  better  power-factors 
may  sometimes  be  obtained.  On  the  other  hand,  the  power- 
factors  on  which  the  curves  are  based,  are,  in  most  instances, 
already  representative  of  fairly  extreme  proportions  in  this  respect; 
and  few  manufacturers  find  it  commercially  practicable  to  list 
low-speed  motors  with  such  high  power-factors  as  are  indicated 
by  these  curves.  The  purchaser  would  rarely  be  willing  to  pay 
a  price  which  would  leave  any  margin  of  profit  were  these  power- 
factors  provided;  and  consequently  it  is  only  relatively  to  one 
another  that  these  curves  are  of  interest.  They  teach  the  lesson 
that  for  periodicities  of  50  or  60  cycles,  it  must  be  carefully  kept 
in  mind  that,  if  low  speed  induction  motors  are  used,  either  the 
price  paid  must  be  disproportionately  high,  or  else  the  purchaser 
must  be  content  with  motors  of  exceedingly  low  power-factors. 
The  power-factor,  furthermore,  decreases  rapidly  for  a  given  low 
speed,  with  decreasing  rated  output. 

On  the  other  hand,  for  a  25-cycle  supply,  these  considerations 
are  of  decidedly  less  importance.  With  a  clear  recognition  of 
this  state  of  affairs,  a  power  user  desiring  low-speed  motors 
for  a  60-cycle  circuit  will  do  well  to  take  into  careful  considera- 
tion the  alternative  of  employing  synchronous  motors.  If  he 
resorts  to  this  alternative  he  can  maintain  his  power-factor  at 
unity,  irrespective  of  the  rated  speed  of  his  motors;  but  he  must 
put  up  with  the  slight  additional  complication  of  providing  for  a 
rotor  which,  in  addition  to  the  squirrel-cage  winding,  is  also 
equipped  with  field  windings  excited  through  brushes  and  slip- 
rings  from  a  source  of  continuous  electricity.  The  precise 
circumstances  of  any  particular  case  will  require  to  be  considered, 
in  order  to  decide  whether  or  not  this  alternative  is  preferable. 

While  it  has  long  been  recognized  that  the  inherent  simplicity 
and  robustness  of  the  squirrel-cage  induction  motor  constitute 
features  of  very  great  importance  and  justify  the  wide  use  of 
such  motors,  nevertheless  we  must  not  overlook  the  fact  that 


204       POLYPHASE  GENERATORS  AND  MOTORS 

induction  motors  are  less  satisfactory  the  lower  the  rated  speed. 
The  chief  disadvantage  of  a  low-speed  induction  motor  is  its  low 
power-factor  as  shown  above.  Let  us  take  the  case  of  a  100-h.p., 
60-cycle  motor.  The  design  for  a  rated  speed  of  1800  r.p.m. 
will  (see  Fig.  109)  have  a  power-factor  of  93  per  cent,  whereas 
the  design  for  a  100-h.p.,  60-cycle  motor  for  one-tenth  of  this 
speed,  i.e-.,  for  a  speed  of  only  180  r.p.m.,  will  have  a  power- 
factor  of  only  some  76  per  cent.  The  power-factors  given 
above  correspond  to  rated  load.  For  light  loads,  the  inferiority 
of  the  low-speed  motor  as  regards  low  power-factor  is  much 
greater,  and  this  circumstance  further  accentuates  the  objection 
to  the  use  of  such  a  motor. 


LOO 


0.90 


0.80 


0.70 


0.60 

100     200     300     400     500      600      700     800      900    1000   1100    1200    1300   1400^1500 
Hated  Speed  in_R.P.M. 

FIG.  108. — Curves  of  Power-factors  of  25-cycle  Induction  Motors  of  5  H.P., 
50  H.P.,  and  500  H.P.  Rated  Output,  Plotted  with  Rated  Speeds  as 
Abscissae. 


For  a  long  time  there  has  existed  a  general  impression  that  a 
synchronous  motor  could  not  be  so  constructed  as  to  provide 
much  starting  torque.  Otherwise  it  would  probably  have  been 
realized  that,  for  low  speeds,  it  would  often  be  desirable  to  give 
the  preference  to  the  synchronous  type.  For  with  the  synchro- 
nous type,  the  field  excitation  can  be  so  adjusted  that  the  power- 
factor  shall  be  unity;  indeed  there  is  no  objection  to  running 
with  over-excitation  and  reducing  the  power-factor  again  below 
unity,  thus  occasioning  a  consumption  of  leading  current  by  the 
synchronous  motor. 

By  the  judicious  admixture  (on  a  single  supply  system)  of 


SYNCHRONOUS  MOTORS  vs.  INDUCTION  MOTORS    205 

high-speed  induction  motors  consuming  a  slightly-lagging  current 
and  low-speed  synchronous  motors  consuming  a  slightly  leading 
current,  it  is  readily  feasible  to  operate  the  system  at  practically 
unity  power-factor;  and  thereby  to  obtain,  in  the  generating 
station  and  on  the  transmission  line,  the  advantages  usually 
accruing  to  operation  under  this  condition. 

Now  the  point  which  is  beginning  to  be  realized,  and  which 
it  is  important  at  this  juncture  to  emphasize,  is  that  the  synchro- 
nous motor,  instead  of  being  of  inferior  capacity  as  regards  the 
provision  of  good  starting  torque,  has,  on  the  contrary,  certain 
inherent  attributes  rendering  it  entirely  feasible  to  equip  it  for 
much  more  liberal  starting  torque  than  can  be  provided  by 


0.60 

100  200  300  400  500   600  700  800  900  1000  1100  1200  1300  1400  1500 

Bated  Speed  in  R. P.M. 

FIG.  109. — Curves  of  Power-factors  of  50-cycle  Induction  Motors  of  5  H.P., 
50  H.P.,  and  500  H.P.  Rated  Output,  Plotted  with  Rated  Speeds  as 
Abscissae. 

efficient  squirrel-cage  induction  motors.  The  word  "  efficient  " 
has  been  emphasized  in  the  preceding  statement  and  for  the 
following  reason:  by  supplying  an  induction  motor  with  a 
squirrel-cage  system  composed  of  conductors  of  sufficiently 
small  cross-section,  and  consequently  sufficiently-high  resistance, 
any  amount  of  starting  torque  which  is  likely  to  be  required, 
can  be  provided.  Unfortunately,  however,  the  high -resistance 
squirrel  cage  is  inherently  associated  with  a  large  PR  loss  in 
the  rotor  when  the  motor  is  carrying  its  load.  This  large  PR 
loss  not  only  occasions  great  "  slip  "  but  also  occasions  very  low 
efficiency.  Furthermore,  owing  to  this  very  low  efficiency, 


206       POLYPHASE  GENERATORS  AND  MOTORS 

the  heating  of  the  motor  would  be  very  great  were  it  not  that 
such  a  motor  is  rated  down  to  a  capacity  far  below  the  capacity 
at  which  it  could  be  rated  were  it  supplied  with  a  low  resistance 
(i.e.,  low  starting  torque)  squirrel-cage  system. 

But  when  we  turn  to  the  consideration  of  the  synchronous 
motor,  we  note  the  fundamental  difference,  that  the  squirrel-cage 
winding  with  which  we  provide  the  rotor,  is  only  active  during 
starting,  and  during  running  up  toward  synchronous  speed. 
When  the  motor  has  run  up  as  far  toward  synchronous  speed 
as  can  be  brought  about  by  the  torque  supplied  by  its  squirrel  cage, 
excitation  is  applied  to  the  field  windings  and  the  rotor  pulls  in 
to  synchronous  speed.  So  soon  as  synchronism  has  been  brought 
about,  the  squirrel-cage  system  is  relieved  of  all  further  duty; 
and  it  is  consequently  immaterial  whether  it  is  designed  for  high 
or  for  low  resistance. 

Consequently  with  the  synchronous  motor  we  are  completely 
free  from  the  limitations  which  embarrass  us  in  designing  high- 
torque  induction  motors.  In  the  case  of  the  synchronous  motor, 
we  can  provide  any  reasonable  amount  of  torque  by  making  the 
squirrel-cage  system  of  sufficiently  high  resistance.  One  difficulty, 
however,  presents  itself:  while,  at  the  instant  of  starting,  we  may 
desire  very  high  torque  and  may  provide  it  by  a  high-resistance 
squirrel  cage,  the  higher  the  resistance  the  more  will  the  speed, 
up  to  which  the  rotor  will  be  brought  by  the  torque  of  the  squirrel 
cage,  fall  short  of  synchronous  speed.  It  would  in  fact,  be  desirable 
that  as  the  rotor  acquires  speed  the  squirrel  cage  should  gradually 
be  transformed  from  one  of  high  resistance  to  one  of  low  resistance. 
If  we  could  accomplish  the  result  that  the  squirrel  cage  should, 
at  the  moment  the  motor  starts  from  rest,  have  a  high  resistance, 
and  if  it  could  be  arranged  that  this  resistance  should  gradually 
die  away  to  an  exceedingly  low  resistance  as  the  motor  speeds  up, 
then  the  motor  would  gradually  run  up  to  practically  synchronous 
speed  and  would  furnish  ample  torque  throughout  the  range  from 
zero  to  synchronous  speed. 

We  can  provide  precisely  this  arrangement  if,  instead  of  making 
the  end  rings  of  the  squirrel  cage,  of  copper  or  brass  or  other  non- 
magnetic material,  we  employ  instead,  end  rings  of  magnetic  mate- 
rial, such  as  wrought  iron,  mild  steel,  cast  iron  or  some  magnetic 
alloy.  Let  us  consider  the  reason  why  this  arrangement  should 
produce  the  result  indicated.  Just  before  the  motor  starts,  the 


SYNCHRONOUS  MOTORS  vs.  INDUCTION  MOTORS    207 

current?  induced  in  the  squirrel-cage  system  are  of  the  full  perio- 
dicity of  the  supply.  In  the  end  rings,  these  currents  will,  with 
usual  proportions,  be  very  large  in  amount;  and  since  the  currents 
are  alternating  and  since  the  material  of  the  end  rings  is  magnetic, 
there  will  be  a  very  strong  tendency,  in  virtue  of  the  well-known 
phenomenon  generally  described  as  "  skin  effect,"  to  confine  the 
current  to  the  immediate  neighborhood  of  the  surface  of  the 
end  rings.  The  current  will  be  unable  to  make  use  of  the  full 
cross-section  of  the  end  rings;  and  consequently,  even  though 
the  end  rings  may  be  proportioned  with  very  liberal  cross-section, 
the  net  result  will,  at  starting,  be  the  same  as  if  the  end  rings  were 
of  high  resistance.  But  as  the  motor  speeds  up,  the  periodicity 
of  the  currents  in  the  squirrel  cage  decreases,  until,  at  synchro- 
nism, the  periodicity  would  be  zero  and  there  would  be  no  "  skin 
effect."  In  view  of  these  explanations,  it  is  obvious  that  the 
impedance  of  the  end  rings  will  gradually  decrease  from  a  high 
value  at  starting  to  a  low  value  at  synchronism. 

Now  we  are  more  free  to  make  use  of  this  phenomenon  in  the 
case  of  synchronous  motors  than  in  the  case  of  induction  motors, 
for  as  previously  stated,  when  the  synchronous  motor  is  run  at  full 
speed,  the  squirrel  cage  is  utterly  inactive  (except  in  serving  to 
minimize  "  surging  "  and  to  decrease  "  ripple  "  losses);  whereas 
the  induction  motor's  squirrel  cage  is  always  carrying  alternating 
current  (even  though  of  low  periodicity);  and  this  alternating 
current  flowing  through  end  rings  of  magnetic  material,  occasions 
a  lower  power-factor  than  would  be  the  case  with  the  equivalent 
squirrel-cage  motor  with  end  rings  of  non-magnetic  material. 
Even  in  the  case  of  induction  motors,  excellent  use  can  be  made 
of  constructions  with  end  rings  of  magnetic  material,  in  improving 
the  starting  torque,  but  there  is  unavoidably  at  least  a  little  sacri- 
fice in  power-factor  during  normal  running. 

It  should  now  be  clear  that  there  is  a  legitimate  and  wide 
field  for  low-speed  synchronous  motors  and  that  these  motors  will 
be  superior  to  low-speed  induction  motors,  in  that,  while  the  former 
can  be  operated  at  unity  power-factor,  or  even  with  leading  cur- 
rent if  desired,  the  latter  will  unavoidably  have  very  low  power- 
factors.  Furthermore  these  low-speed  synchronous  motors, 
instead  of  being  in  any  way  inferior  to  induction  motors  with 
respect  to  starting  torque,  have  attributes  permitting  of  providing 
them  with  higher  starting  torque  than  can  be  provided  with 


208       POLYPHASE  GENERATORS  AND  MOTORS 

induction  motors,  without  impairing  other  desirable  character- 
istics such  as  low  heating  and  high  efficiency. 

Of  course  there  always  remains  the  disadvantage  of  requiring 
a  supply  of  continuous  electricity  for  the  excitation  of  the  field 
magnets.  Cases  will  arise  where  this  disadvantage  is  sufficient 
to  render  it  preferable,  even  for  low-speed  work,  to  employ 


1        2        3        4        5        6         7        8        9       10      11      12      13       14 
Ratio  of  Apparent  to  True  Resistance 

FIG.  110. — Robinson's  Curves  for  Skin  Effect  in   Machine-steel  Bar  with  a 
Cross-section  of  25X25  mm. 


induction  motors,  but  in  the  majority  of  cases  where  polyphase 
motors  must  operate  at  very  low  speeds,  it  would  appear  that 
synchronous  motors  are  preferable. 

In  the  curves  in  Figs.  110  and  111  are  plotted  the  results  of 
some  interesting  tests  which  have  recently  been  made  by  Mr. 
L.  T.  Robinson,  on  "  skin  effect  "  in  machine  steel  bars.  By 


SYNCHRONOUS  MOTORS  vs.  INDUCTION  MOTORS    209 

means  of  these  and  similar  data,  and  by  applying  to  the  design 
of  the  synchronous  motor  the  ample  experience  which  has  been 
acquired  in  the  design  of  induction  motors,  the  preparation  of 
a  design  for  a  synchronous  motor  for  stipulated  characteristics 
as  regards  torque,  presents  no  difficulties.  The  smooth-core 
type  of  field  with  distributed  excitation  is  to  be  preferred  to  the 
salient-pole  type. 

We  have  considered  the  relative  inappropriateness  of  the 
induction  motor  for  low-speed  applications.  Conversely  it  is 
a  particularly  excellent  machine  for  high  speeds.  Its  power- 


10 


20 


80 


90   -    100 


50          60          70 
Periodicity  in  Cycles  per  Second 

FIG.  111. — Robinson's  Curves  for  Skin  Effect  in  Machine-steel  Bar  with  a 
Cross-section  of  25X25  mm. 

factor  is  higher  the  higher  the  rated  speed;  and  when  we  come 
to  very  high  speeds  we  obtain,  in  motors  of  large  capacity,  full- 
load  power-factors  in  excess  of  95  per  cent.  In  such  instances 
the  simplicity  of  squirrel-cage  induction  motors  should  frequently 
lead  to  their  use  in  preference  to  synchronous  motors. 

Reasoning  along  similar  lines,  in  the  case  of  generators,  the 
induction  type  offers  advantages  over  the  synchronous  type  in 
many  instances.  It  cannot,  however,  replace  the  synchronous 
generator  even  at  high  speeds,  for  as  explained  in  Chapter  VII,  it 
requires  to  be  run  in  parallel  with  synchronous  generators,  the 


210       POLYPHASE  GENERATORS  AND  MOTORS 

latter  supplying  the  magnetization  for  the  induction  generators 
and  also  supplying  the  lagging  component  of  the  external  load  when 
the  latter 's  power-factor  is  less  than  unity.  Notwithstanding 
these  limitations,  there  is  a  wide  field  for  the  induction  generator; 
and  the  above  indications  may  be  useful  for  guidance  in  showing 
its  appropriateness  in  any  specific  instance. 

It  will  now  be  agreed  that  the  properties  of  synchronous  motors 
lend  themselves  admirably  to  the  provision  of  high  starting  torque. 
In  addition  to  the  means  previously  described  whereby  a  synchro- 
nous motor  may  have  not  only  high  starting  torque  but  may  also 
automatically  run  close  up  to  synchronous  speed,  so  as  to  fall 
quietly  into  synchronism  immediately  upon  the  application  of 
the  field  excitation  from  the  continuous-electricity  source,  there 
is  also  available  a  phenomenon  described  by  Mr.  A.  B.  Field,  in  a 
paper  entitled  "  Eddy  Currents  in  Large  Slot- Wound  Con- 
ductors/' presented  in  June,  1905,  before  the  American  Institute 
of  Electrical  Engineers  (Vol.  24,  p.  761). 

Mr.  A.  B.  Field  analyzed  the  manner  in  which  the  apparent 
resistance  of  slot-embedded  conductors  varies  with  the  periodicity. 
Subsequent  investigations  show  that  the  Field  phenomenon, 
while  harmful  in  stator  windings  exposed  constantly  to  the  full 
periodicity  of  the  supply  system,  may  be  employed  to  consider- 
able advantage  in  the  proportioning  of  the  conductors  of  the  slot 
portion  of  a  squirrel-cage  system.  Both  in  synchronous  and  in 
induction  motors,  this  is  an  important  step.  Take,  for  instance, 
the  case  of  a  60-cycle  induction  motor.  At  starting,  the  periodicity 
of  the  currents  in  the  squirrel-cage  system  is  60  cycles,  and  the 
Field  effect  may,  with  properly-proportioned  conductors,  be 
sufficient  to  occasion  an  apparent  resistance  very  much  greater 
than  the  true  resistance.  The  motor  thus  starts  with  very  much 
more  torque  per  ampere,  than  were  the  Field  phenomenon  absent. 
As  the  motor  speeds  up,  the  periodicity  decreases  and  the  Field 
effect  dies  out,  the  apparent  resistance  of  the  squirrel-cage  grad- 
ually dying  down  to  the  true  resistance.  Thus,  whereas  the  full- 
load  running  slip  of  an  ordinary  squirrel-cage  motor  must  be  high, 
its  heating  high  and  its  efficiency  low,  if  it  is  to  develop  high  start- 
ing torque,  we  may,  by  using  the  Field  effect,  construct  high-start- 
ing torque  motors  with  low  slip  and  high  efficiency. 

In  Fig.  112  are  shown  respectively  the  rough  characteristic 
shape  of  a  curve  in  which  torque  is  plotted  as  a  function  of 


SYNCHRONOUS  MOTORS  vs.  INDUCTION  MOTORS    211 

the  speed  for  :  A,  a  permanently  low-resistance  squirrel-cage 
motor;  B,  a  permanently  high-resistance  squirrel-cage  motor; 
and,  C,  for  a  squirrel-cage  motor  in  which  the  apparent  resist- 
ance gradually  changes  from  a  high  to  a  low  value  as  the  motor 
runs  up  from  rest  to  synchronism.  As  regards  the  application 
of  these  curves  to  a  synchronous  motor,  it  will  be  seen  that,  in  the 
first  case,  (Curve  A),  the  starting  torque  is  rather  low;  but  that 
the  torque  increases,  passes  through  a  maximum,  and  falls  slowly, 
remaining  quite  high  until  the  speed  is  close  to  synchronism. 
This  motor,  while  unsatisfactory  at  starting,  has  the  property 


0        10       20       30      40       50       60       70       80       90     100 
Speed  in  Percent  of  Synchronous  Speed 

FIG.  112. — Curves  Contrasting  Three  Alternative  Squirrel-cage  Constructions 
for  a  Synchronous  Motor. 


of  pulling  easily  into  synchronism  on  the  application  of  the 
excitation  fr.m  the  continuous-electricity  source.  In  the  second 
case,  (Curve  J5),  while  the  synchronous  motor  starts  with  high 
torque,  the  torque  falls  away  much  more  rapidly;  and  when  the 
torque  has  fallen  to  the  value  necessary  to  overcome  the  friction 
of  the  motor,  the  speed  is  several  per  cent  below  synchronous 
speed,  and  the  application  of  the  continuous  excitation,  if  it 
suffices  to  pull  the  rotor  into  synchronism,  will  do  so  only  at  the 
cost  of  an  abrupt  and  considerable  instantaneous  drain  of  power 
from  the  line.  In  the  third  case,  (Curve  C),  there  are  present 


212       POLYPHASE  GENERATORS  AND  MOTORS 


the  good  attributes  of  the  first  two  cases,  the  bad  attributes 
being  completely  eliminated. 

It  is  thought  that  with  the  explanations  furnished  in  this 
Chapter,  the  user  will  be  assisted  in  determining,  in  any  particular 
case,  whether  it  is  more  desirable  to  take  advantage  of  the 
extreme  simplicity  and  toughness  of  the  low-speed  squirrel-cage 
induction  motor,  notwithstanding  its  poor  power-factor,  or 
whether  he  should  employ  the  slightly  more  complicated  syn- 
chronous motor  in  order  to  have  the  advantage  of  high 
power-factor. 

Another  way  of  dealing  with  the  situation,  which  in  certain 
cases  of  a  low-speed  drive  is  preferable,  is  to  employ  a  high- 
speed induction  motor  (which  will  consequently  have  a  high 
power-factor),  and  to  gear  it  down  to  the  low-speed  load.  Thus, 
from  the  curves  in  Fig.  109,  we  see  that,  if  we  require  to  drive 
a  load  at  200  r.p.m.,  a  50-h.p.  motor  will  have  a  power-factor  of 
only  about  0.70  ;  whereas,  if  the  motor  were  to  drive  the  load 
through  5  to  1  gearing,  the  motor's  own  speed  would  be  1000 
r.p.m.,  and  its  power-factor  would  be  over  0.90.  Not  only  would 
the  induction  motor  be  characterized  by  a  20  per  cent  higher 
power-factor  at  full  load,  but  its  efficiency  would  be  higher  and  it 
would  be  much  smaller,  lighter  and  cheaper.  At  half  load  the 
advantage  of  the  high-speed  motor  in  respect  to  power-factor  is 
still  more  striking,  the  two  values  being  of  the  following  order: 


Rated  Speed. 

Power-factor  at 
Half  Load. 

200  r.p.m. 
1000  r.p.m. 

0.50 

0.78 

CHAPTER  VII 
THE  INDUCTION  GENERATOR 

FOR  the  very  high  speeds  necessary  in  order  to  obtain  the  best 
economy  from  steam  turbines,  the  design  of  synchronous  genera- 
tors is  attended  with  grave  difficulties.  The  two  leading  difficul- 
ties relate,  first  to  obtaining  a  sound  mechanical  construction  at 
these  high  speeds,  and,  secondly,  to  the  provision  of  a  field  winding 
which  shall  operate  at  a  permissibly  low  temperature. 

It  is  very  difficult  to  provide  adequate  ventilation  for  a  rotating 
field  magnet  of  small  diameter  and  great  length.  The  result 
has  been  that  for  so  high  a  speed  as  3600  r.p.m.,  the  largest 
synchronous  generators  which  have  yet  given  thoroughly  satis- 
factory results  are  only  capable  of  a  sustained  output  of  some 
5000  kva.  Even  at  this  output  the  design  has  very  undesirable 
proportions  and  the  temperature  attained  by  the  field  winding 
is  undesirably  high. 

But  an  Induction  generator  is  exempt  from  the  worst  of  these 
difficulties.  The  conducting  system  carried  by  its  rotor  consists  in 
a  simple  squirrel  cage,  the  most  rugged  construction  conceivable. 
Thus  from  the  mechanical  standpoint  the  induction  generator 
is  an  excellent  machine.  Furthermore,  the  squirrel  cage  can  be 
so  proportioned  that  the  full-load  PR  loss  is  exceedingly  low. 
Hence  the  rotor  will  run  cool,  and  is,  in  this  respect,  in  striking 
contrast  with  the  rotor  of  a  high-speed  synchronous  generator. 

Notwithstanding  these  satisfactory  attributes,  induction  gen- 
erators are  rarely  employed.  The  chief  obstacle  to  their  extensive 
use  relates  to  the  limitation  that  they  must  be  operated  in  parallel 
with  synchronous  apparatus.  Induction  generators  are  dependent 
for  their  excitation  upon  lagging  current  drawn  from  synchronous 
generators,  or  leading  current  delivered  to  synchronous  motors 
connected  to  the  network  into  which  the  induction  generators 
deliver  their  electricity. 

213 


214       POLYPHASE  GENERATORS  AND  MOTORS 

The  practical  aspects  of  the  theory  of  the  induction  generator 
have  been  considered  in  a  paper  (entitled  "  The  Squirrel-cage 
Induction  Generator  ")  written  in  collaboration  by  Mr.  Edgar 
Knowlton  and  the  present  author.  This  paper  was  presented 
on  June  28,  1912,  at  the  29th  Annual  Convention  of  the  American 
Institute  of  Electrical  Engineers  at  Boston.  It  would  not 
be  appropriate  to  reproduce  the  descriptions  and  explanations 
in  that  paper,  since  the  purpose  of  the  present  treatise  is  to  set 
forth  the  fundamental  methods  of  procedure  employed  in  design- 
ing machinery.  Some  brief  extracts  from  the  paper  in  question 
are  given  in  a  later  part  of  this  chapter,  and  the  reader  will  find 
it  profitable  to  consult  the  original  paper  in  amplifying  his 
knowledge  of  the  practical  aspects  of  the  theory  of  the  induction 
generator. 

Although  the  induction  generator  is  chiefly  suitable  for  large 
outputs,  we  can  nevertheless,  in  explaining  designing  methods, 
employ  as  an  illustrative  example  the  case  of  adapting  to  the 
purposes  of  an  induction  generator,  the  200-h.p.  squirrel-cage 
induction  motor  which  we  worked  out  in  Chapter  IV. 

The  full-load  efficiency  has  (see  p.  178)  been  ascertained  to  be 
93.0  per  cent.  Consequently  at  full  load  the  input  is  ; 

200X746 


Let  us,  in  the  first  instance,  assume  that  the  machine  is  suitable 
for  a  rated  output  of  160  kw.  when  operated  as  an  induction 
generator.  Since,  when  employed  for  this  purpose,  there  is  no 
longer  any  necessity  for  taking  into  consideration  any  questions 
relating  to  starting  torque,  let  us  reduce  the  rotor  PR  loss  by 
widening  the  rotor  face  conductors.  In  the  original  design  of 
the  induction  motor  the  slip  corresponding  to  200  h.p.  was  2.0 
per  cent.  This  slip  corresponds  to  a  rotor  PR  loss  of  3140  watts. 
This  loss  is  made  up  of  two  components,  which  are  2512  watts 
in  the  face  conductors  and  628  watts  in  the  end  rings. 

The  tooth  density  in  the  rotor  is  needlessly  low,  other  con- 
siderations, not  entering  into  induction  generator  design  having 
determined  the  width  of  the  slot.  There  is  now  nothing  to  pre- 
vent doubling  the  width  of  the  rotor  conductors.  This  gives 


THE  INDUCTION  GENERATOR  215 

for  the  dimensions  of  these  conductors  a  depth  of  54  mm.  and 
a  width  of  9.6  mm. 

This  alteration  will  reduce  the  rotor  loss  at  160  kw.  output  to 


——  +628  =  1884  watts. 


The  slip  will  now  be: 


1^X2.0  =  1.2  per  cent. 


In  the  design  of  an  induction  generator,  the  slip  should  be 
made  as  small  as  practicable,  in  order  to  have  a  minimum  rotor 
loss  and  consequently  the  highest  practicable  efficiency  and  low 
heating.  In  large  induction  generators  there  is  rarely  any 
difficulty  in  bringing  the  slip  down  to  a  small  fraction  of  1  per  cent. 
The  slip  of  the  7500-kw.  750-r.p.m.  induction  generators  in  the 
Interborough  Rapid  Transit  Co.'s  59th  Street  Electricity  Supply 
Station  in  New  York,  is  only  about  three-tenths  of  1  per  cent  at 
rated  load. 

Since  we  have  reduced  the  squirrel-cage  loss  by : 

(2.0-1.2  =  )0.8 

per  cent,  the  efficiency  of  our  160-kw.  induction  generator  will  be 
(93.0+0.8  =  )93.8  per  cent. 

It  is  not  this  small  increase  in  efficiency  which  is  particularly 
to  be  desired,  but  the  decreased  heating.  The  total  losses  are 
decreased  from : 

(100.0-93.0  =  )7.0  per  cent  of  the  input 
to 

(100.0-93.8  =  )6.2  per  cent  of  the  input. 


216       POLYPHASE  GENERATORS  AND  MOTORS 

If,  for  a  rough  consideration  of  the  case,  we  take  the  heating 
to  be  proportional  to  the  total  loss,  this  result  would  justify  us 
in  giving  careful  consideration  to  the  question  of  the  feasibility 
of  rating  up  the  machine  in  about  the  ratio  of  : 

6.2  :  7.0. 
This  would  bring  the  rated  output  up  to 


But  before  finally  determining  upon  increasing  the  rating, 
it  would  be  necessary  to  examine  into  the  question  of  the  heating 
of  the  individual  parts,  since  in  decreasing  the  rotor  heating  it 
does  not  necessarily  follow  that  it  is  expedient  to  increase  the 
stator  heating  to  the  extent  of  the  amount  of  the  decreased  loss 
in  the  rotor.  On  the  other  hand,  in  the  case  of  very  high  speed 
generators,  the  heating  of  the  rotor  conductors  will,  practically, 
always  constitute  the  limit,  owing  to  the  difficulty  of  circulating 
air  through  a  rotor  of  small  diameter  and  great  length.  Con- 
sequently in  the  case  of  very  high  speed  generators  the  increased 
rating  rendered  practicable  by  the  substitution  of  a  low-loss 
squirrel-cage  rotor  for  a  rotor  excited  with  continuous  electricity, 
will  often  be  much  greater  than  in  the  inverse  ratio  of  the  respec- 
tive total  losses  for  the  two  cases. 

The  author's  present  object  is,  however,  to  point  out  that  the 
induction  generator  has  inherent  characteristics  usually  permit- 
ting of  assigning  to  it  a  materially  higher  rating  than  that  which 
is  appropriate  when  the  same  frame  is  employed  in  the  con- 
struction of  an  induction  motor. 

Since  the  slip  is  1.2  per  cent,  the  full-load  speed  is 

(1.012X500  =  )  506r.p.m. 

The  Derivation  of  a  Design  for  an  Induction  Generator  from 
a  Design  for  a  Synchronous  Generator.  Let  us  now  evolve  an 
induction  generator  from  the  2500-kva.  synchronous  generator  for 
which  the  calculations  have  been  carried  through  in  Chapter  II. 


THE  INDUCTION  GENERATOR 


217 


This  machine  was  designed  for  the  supply  of  25-cycle  electricity. 
It  had  8  poles  and  operated  at  a 
speed  of  375  r.p.m.  Let  us  first 
plan  not  to  alter  the  stator  in  any 
respect  except  to  employ  a  nearly- 
closed  slot  of  the  dimensions  indi- 
cated in  Fig.  113.  This  alteration 
from  the  slot  proportions  employed  in 
the  synchronous  generator  is  necessary 
in  order  to  avoid  parasitic  losses 
when  the  machine  is  loaded.  The 
necessity  arises  from  the  circumstance 
that,  unlike  a  synchronous  genera- 
tor, an  induction  generator  must  be 
designed  with  a  very  small  air-gap. 
Otherwise  it  would  have  an  unde- 
sirably-low power-factor  and  would 
require  the  supply  of  too  considerable  a 
magnetizing  current  from  the  syn- 
chronous apparatus  with  which  it  operates  in  parallel.  In  the  case 
with  which  we  are  dealing,  we  may  employ  an  air-gap  depth  of 

—       only  2  mm.    Thus  we  have: 


H 


FIG.  113.— Stator  Slot  for 
Induction  Generator. 


A -0.20. 


-25-mm- 


FIG.  114. — Slot  for  Rotor  of  Induction 
Generator. 


The  stator  has  120  slots. 
Let  us  supply  the  rotor  with 
106  slots.  Each  rotor  slot  may 
be  made  50  mm.  deep  and  25 
mm.  wide.  The  slot  is  closed, 
as  shown  in  Fig.  114,  by  a  solid 
steel  wedge  with  a  depth  of  10 
mm.  The  conductor  is  unin- 
sulated and  is  40  mm.  deep  by 
25  mm.  wide,  its  cross-section 
thus  being: 

4X2.5  =  10sq.cm. 
(120X10  =  )  1200     stator    conductors    and 


Since     there     are 
(106X1  =  )106  rotor  conductors,  the  ratio  of  transformation  is: 

1200 


106 


=  11.3. 


218       POLYPHASE  GENERATORS  AND  MOTORS 

The  energy  component  of  the  full-load  current  in  the  stator 
winding  is  120  amperes.  Neglecting  the  magnetizing  component 
of  the  stator  current,  we  may  obtain  a  rough,  but  sufficient, 
approximation  to  the  value  of  the  full-load  current  per  rotor 
conductor.  This  is  : 

(11.  3X120  =  )  1360  amperes. 

The  current  density  in  the  rotor  conductors  is  thus: 

1360 

—  —  =  136  amp.  per  sq.cm. 

The  gross  core  length  is: 


g  =        cm. 

Allowing  8  cm.  for  the  projections  at  each  end,  the  total  length 
of  each  rotor  conductor  is 

(118+2X8  =  )134cm. 
The  aggregate  length  of  the  106  rotor  conductors  is 

(106X134  =  )  14  200  cm. 
The  corresponding  resistance,  at  60°  Cent,  is: 


The  PR  loss  in  the  rotor  face  conductors  at  full  load  is 

13602  X  0.00284  =  5250  watts. 

At  full  load,  the  current  in  each  end  ring  is  (see  p. 
X  1360  =  5700  amp. 


THE  INDUCTION  GENERATOR  219 

Let  us  give  each  end  ring  a  cross-section  of  40  sq.cm. 

The  mean  diameter  of  an  end  ring  must  be  a  little  less  than 
D,  i.e.,  a  little  less  than  178  cm.  Let  us  take  the  mean  diameter 
of  the  end  ring  as  165  (  m. 

The  resistance  of  a  conductor  equal  to  the  aggregate  of  the 
developed  length  of  the  two  end  rings  is,  at  60°  Cent.  : 


The  full-load  loss  in  the  two  end  rings  is  : 

57002X  0.000052  =  1680  watts. 
The  total  loss  in  the  squirrel  cage,  at  full  load  is  : 
5250+1680  =  6930  watts. 

If  our  induction  generator  were  to  be  rated  at  2500  kw.,  this 
rotor  loss  would  be  : 

6930X100     ftOQ 
2100^00  =0'28perCent- 

We  have  seen  (p.  90)  that  the  loss  in  the  rotor  of  our  2500 
kva.  synchronous  generator  is  (at  a  power-factor  of  0.90  and 
consequently  an  output  of  2250  kw.)  15  500  watts. 

Thus  the  efficiency  is  considerably  greater  in  the  case  of  the 
induction  generator  rating  of  2500  kw.  Indeed,  since  the  hottest 
part  of  the  synchronous  generator  is  its  rotor,  we  can  easily  rate 
up  the  machine,  when  re-modelled  as  an  induction  generator, 
to  3000  kw,  the  slip  then  being: 


=00.33  per  cent. 


Since,  however,  the  air-gap  is  now  so  small  as  to  be  of  but  little 
service  as  a  channel  through  which  to  circulate  cooling  air,  the 
preferable  design  for  the  induction  generator  would  consist  in 
a  modification  in  which,  instead  of  employing  numerous  vertical 


220       POLYPHASE  GENERATORS  AND  MOTORS 

ventilating  ducts,  the  air  is  circulated  through  120  longitudinal 
channels,  one  just  below  each  stator  slot.  Two  other  methods 
of  air  circulation  which  have  been  employed  on  the  Continent 


Fio.  115. — A  Method  of  Ventilation  Suitable  for  an  Induction  Generator. 

of  Europe  for  synchronous  generators  and  are  especially  appro- 
priate for  induction  generators,  are  shown  in  Figs.  115  and  116. 
In  that  indicated  in  Fig.  115,  the  air  from  the  fans  on  the  ends  of 
the  rotor  is  passed  to  a  chamber  at  the  external  surface  of  the 


THE  INDUCTION  GENERATOR 


221 


armature  core.  This  chamber  opens  into  air  ducts  in  a  plane 
at  right  angles  to  the  shaft.  Suitably  shaped  space  blocks  lead 
the  air,  in  a  tangential  direction,  to  axial  ducts  just  back  of  the 
stator  slots.  The  air  then  flows  axially  through  one  section,  to 
the  next  air  duct,  and  then  outwardly  in  a  tangential  direction 
to  a  chamber  at  the  outer  surface  of  the  core.  This  chamber  is 
adjacent  to  the  one  first  mentioned  and  leads  to  the  exit  from 
the  stator  frame.  Looking  along  the  axis  of  the  shaft,  the  air 
flows  in  a  V-shaped  path,  the  axial  duct  back  of  the  stator  slots 


FIG.  116. — An  Alternative  Method  of  Ventilation  Suitable  for  an  Induction 

Generator. 

being  at  the  apex.     Thus  the  two  legs  of  the  V  are  separated 
axially  by  a  single  armature  section. 

The  other  method  (Fig.  116)  which  is  also  independent  of  the 
radial  depth  of  the  air-gap,  consists  in  dividing  the  stator  frame 
into  cylindrical  chambers  placed  side  by  side.  The  air  is  forced 
into  a  chamber  from  which  it  first  passes  radially  toward  the  shaft, 
then  axially  to  adjacent  air  ducts,  and  finally  outwardly  to  a 
chamber  alongside  the  one  first  mentioned.  This  last  chamber 
communicates  with  the  outer  air. 


222       POLYPHASE  GENERATORS  AND  MOTORS 

Since  the  loss  in  the  squirrel  cage  is  0.33  of  1  per  cent,  in  the 
case  of  this  induction  generator,  the  speed  at  rated  load  will  be 
1.0033X375  =  376.2  r.p.m. 

For  the  2-mm.  air-gap  which  we  are  now  employing,  the 
mmf.  calculations  for  the  phase  pressure  of  6950  volts,  may 
(without  attempting  to  arrive  at  a  needlessly  exact  result)  be 
estimated  as  follows: 

Air-gap 1300  ats. 

Teeth 700  " 

Stator  core 300   " 

Rotor  core.  .  300  " 


Total  mmf.  per  pole  .............  =2600  ats. 

Thus  each  phase  must  supply  1300  ats.     There  are  25  turns 
per  pole  per  phase.     Thus  the  no-load  magnetizing  current  is: 

1300 

=  37  amperes. 


Without  going  into  the  estimation  of  the  circle  factor  it  is 
evident  that  the  wattless  component  for  full  load  of  3000.  kw. 
will  be  a  matter  of  some  45  amperes. 

The  current  output  at  full  load  of  3000  kw.  is: 

3  000  000 

=  144  amperes. 


The  total  current  in  each  stator  winding)  at  full  load  is: 


\/452+1442  =  151  amperes. 
The  power-factor  at  full  load  is: 

144 


It  must  be  remembered  that  this  design  is  for  the  moderate 
speed  of  375  r.p.m.     For  high-speed  designs   (say  1500  r.p.m. 


THE  INDUCTION  GENERATOR 


223 


at  25  cycles)  the  power-factor,  in  large  sizes,  may  readily  be 
brought  up  to  0.97. 

In  the  paper  to  which  reference  has  already  been  made 
(Hobart  and  Knowlton's  "  The  Squirrel-cage  Induction  Genera- 
tor "),  mention  is  made  of  a  comparative  study  which  has  been 
carried  out  for  two  60-cycle,  3600  r.p.m.  designs,  one  for  a  2500-kw. 
synchronous  generator  and  the  other  for  a  2500-kw.  induction 
generator,  both  for  supplying  a  system  at  unity  power-factor. 
The  leading  data  of  these  two  designs  were  as  follows : 


Synchronous 
Generator. 

Induction 
Generator. 

Core  loss  .  .  
Primary  I2R  loss  

30      kw. 
9.5  " 

30      kw. 
10.5  " 

Secondary.  I2R  loss 

65" 

28" 

Windage  

35 

35 

Total  loss,  excluding  friction  of  bear- 
ings 

81       " 

78  3  " 

Efficiency,  excluding  friction  of  bear- 
ings   

Per  cent  slip  at  rated  load  

96  .  9  per  cent 
0 

97      per  cent 
0.11       " 

The  necessity  for  employing  in  induction  generators  a  very 
small  air-gap  renders  it  more  important  than  in  synchronous 
generators  that  the  magnetomotove  force  per  slot  (expressed  in 
ampere-conductors  at  rated  load)  shall  be  small.  This  requires 
(in  the  case  of  large  machines),  the  subdivision  of  the  winding 
amongst  a  larger  number  of  slots  per  pole  than  would  be  sufficient 
for  a  synchronous  machine  of  the  same  rating.  The  subdivision 
of  the  winding  in  many  slots  increases  the  desirability  of  employ- 
ing a  low  pressure  and  renders  it  especially  advantageous  to  adopt 
the  plan  of  stepping-up  to  the  line  pressure  through  a  compensator. 
The  most  economical  ratio  of  transformation  for  the  compensator 
is  2:1.  Consequently  if  a  large  induction  generator  is  to  supply 
a  12,000-volt  system,  it  should  be  wound  for  6000  volts  and 
should  supply  the  system  through  a  2 : 1  compensator. 

In  large  sizes  of  high-speed  induction  generators,  it  may 
be  preferable  to  use  a  rotor  in  wrhich  the  shaft  and  core  are 
cut  from  a  solid  piece.  Such  a  construction  is  not  objection- 
able for  the  rotor  of  an  induction  generator,  for  we  can  design 


224       POLYPHASE  GENERATORS  AND  MOTORS 

it  with  a  slip  at  rated  load,  of,  say,  two-tenths  of  1  per  cent.  This 
corresponds  to  a  rotor  periodicity  of  only  (0.002X60  =  )0.12 
of  a  cycle  per  second,  or  7.2  cycles  per  minute,  and  under  such 
conditions  there  is  no  necessity  for  employing  a  laminated  core. 

On  account  of  the  small  air-gap  of  induction  generators,  the 
value  of  the  critical  speed  of  vibration  is  especially  important. 
If  possible,  the  critical  speed  should  be  at  least  10  per  cent 
above  normal.  If  other  important  reasons  require  employing  a 
critical  speed  below  normal,  it  should  be  considerably  below, 
care  being  taken  that  the  second  critical  speed  is  also  removed 
from  the  normal,  preferably  above  it.  With  such  a  design  the 
rotor  should  have  a  very  careful  running  balance  before  it  is 
placed  in  the  machine.  A  damping  bearing  could  be  used  to 
prevent  the  rubbing  of  the  rotor  and  stator  if  for  any  reason  the 
machine  should  be  subjected  to  abnormal  vibration. 

It  should  be  noted  that  in  cases  where  the  critical  speed  must 
be  below  the  normal  speed,  the  air-gap  cannot  be  so  small  as 
would  be  preferred  from  the  standpoint  of  minimizing  the  magnet- 
izing current.  A  consideration  tending  to  the  use  of  a  shaft  with 
a  critical  speed  below  the  normal  running  speed  relates  to  the 
lower  peripheral  speed  thereby  obtained  at  the  bearings. 


CHAPTER  VIII 

EXAMPLES   FOR  PRACTICE   IN   DESIGNING  POLYPHASE 
GENERATORS  AND  MOTORS 

IN  connection  with  courses  of  lectures  on  the  subject-matter 
of  this  treatise,  the  author  has  had  occasion  to  set  a  number 
of  examination  papers.  In  the  present  chapter  some  of  these 
examination  papers  are  reproduced  and  it  is  believed  that  they 
will  be  of  service  in  acquiring  ability  to  apply  the  designing 
principles  discussed  in  the  course  of  the  preceding  chapters. 

PAPER  NUMBER  I 

1.  The  leading  data  of  a  12-pole,  250-r.p.m.,  25-cycle,  11,000- 
volt,  3000-kva.,  Y-connected,  three-phase  generator  are  as  follows: 

Output  in  kilovolt-amperes 3000 

Number  of  poles 12 

Terminal  pressure 11  000  volts 

Style  of  connection  of  stator  windings .  .  Y 

Current  per  terminal 157  amperes 

Speed  in  r.p.m 250 

Frequency  in  cycles  per  second 25 

Gross  core  length  of  armature  (\g) 100  cm. 

Total  number  of  slots 108 

Conductors  per  slot 10 

(All  conductors  per  phase  are  in  series) . 

The  no-load  saturation  curve  is  given  in  Fig.  117. 

What  is  the  armature  strength  in  ampere-turns  per  pole 
when  the  output  is  157  amperes  per  phase?  What,  at  25  cycles, 
is  the  reactance  of  the  armature  winding  in  ohms  phase?  Plot 

225 


226       POLYPHASE  GENERATORS  AND  MOTORS 

saturation  curves  when   the    generator    is    delivering   full    load 
current  (i.e.,  157  amperes)  for: 

I.  Power-factor  =  1.00 

II.  "          =0.70 

III.  "         =0.20 

For  a  constant  excitation  of   11  250  ampere-turns  per  field 
spool,  what  will  be  the  percentage  drop  in  terminal  pressure  when 


8000 


2000         4000         GOOO         8000       10,000      12,000      14,000      16,000 
Ampere  Turns  per  Pole 

FIG.  117.— No-load  Saturation  Curve  for  the  3000-kva.  Three-phase  Generator 
Described  in  Paper  No.  1. 

the  output  is  increased  from  0  amperes  to  full-load  amperes, 
i.e.,  to  157  amperes: 

I.  At  power-factor  =  1 .00 

II.  "  =0.70 

III.  "  =0.20 


EXAMPLES  FOR  PRACTICE  IN  DESIGNING     227 

For  a  constant  terminal  pressure  of  11  000  volts,  what  will  be 
the  required  increase  in  excitation  in  going  from  0  amperes  out- 
put to  full  load  (i.e.,  157)  amperes  output 

I.  At  power-factor  =  1.00 

II.  "  =0.70 

III.  "  =0.20 

In  calculating  theta  (6)  for  this  generator,  work  from  the  data 
given  on  pp.  45  and  48. 

2.  A  100-h.p.,  12-pole,  500-r.p.m.,  50-cycle,  500-volt,  Y-con- 
nected,  three-phase  squirrel-cage  induction  motor  has  a  no-load 
current  of  23  amperes  and  a  circle  ratio  (a)  of  0.058.  The  core 
loss  is  2200  watts,  and  the  friction  loss  is  1400  watts.  At  rated 
load  the  PR  losses  are 

Stator 2850  watts 

Rotor. 2850      " 

[By  means  of  a  circle  diagram  and  rough  calculations  based 
on  the  above  data,  plot  the  efficiency,  the  power-factor,  and  the 
amperes  input,  all  as  functions  of  the  output,  from  no  load  up 
to  100  per  cent  overload. 

PAPER  NUMBER  II 

1.  In  a  certain  three-phase,  squirrel-cage  induction  motor,  the 
current  per  phase  at  rated  load  is  60  amperes.  The  no-load 
current  is  20  amperes.  The  circle  ratio  (c)  is  0.040.  Construct 
the  circle  diagram  of  this  motor.  What  is  its  power-factor  at 
its  rated  load?  What  is  the  current  input  per  phase  at  the  load 
corresponding  to  the  point  of  maximum  power-factor,  and  what 
is  the  maximum  power-factor?  If  the  "  stand-still "  current 
is  500  amperes  (i.e.,  if  the  current  when  the  full  pressure  is  switched 
on  to  the  motor  when  it  is  at  rest,  is  500  amperes),  ascertain 
graphically  by  the  aid  of  the  circle  diagram  the  power-factor  at 
the  moment  of  starting.  If  the  stator  windings  are  Y-connected 
and  if  the  terminal  pressure  is  1000  volts  (the  pressure  per  phase 

consequently  being  -  — —=577  volts),  what  would  be  the  input 
•v3 


228       POLYPHASE  GENERATORS  AND  MOTORS 

to  the  motor,  in  watts,  at  the  moment  of  starting  under  these 
conditions?  Describe,  without  attempting  to  give  quantitative 
data,  the  means  usually  employed  in  practice  to  start  such  a 
motor  with  much  less  than  the  above  large  amount  of  power  and 
with  much  less  current  than  the  "  stand-still  "  current  given  above. 
2.  A  three-phase,  Y-connected,  squirrel-cage  induction  motor 
has  48  stator  slots  and  12  conductors  per  slot.  The  terminal 
pressure  is  250  volts  (the  pressure  per  phase  consequently  being 
144  volts).  The  motor  has  4  poles  and  its  speed,  at  no  load,  is 
1500  r.p.m.  Estimate  the  magnetic  flux  per  pole.  The  squirrel 
cage  comprises  37  face  conductors,  each  having  a  cross-section 
of  0.63  sq.cm.  and  each  having  a  length  between  end  rings  of  20 
cm.  Each  end  ring  has  a  cross-section  of  2.45  sq.cm.  and  a  mean 
diameter  of  20  cm.  Calculate  the  PR  loss  in  the  squirrel  cage 
when  the  current  in  the  stator  winding  is  17.3  amperes.  If, 
without  making  any  further  alteration  in  the  motor,  the  cross- 
section  of  the  end  rings  is  reduced  to  one-half,  what  will  be  the 
PR  loss  in  the  squirrel  cage  for  this  same  current?  What  general 
effect  will  this  change  have  on  the  "  slip  "  ?  On  the  starting 
torque?  On  the  efficiency?  On  the  heating? 

PAPER  NUMBER  III 

Determine  appropriate  leading  dimensions  and  calculate  as 
much  as  practicable  of  the  following  design: 

A  25-cycle,  250-volt,  three-phase,  squirrel-cage  induction 
motor  for  750  h.p.  and  a  synchronous  speed  of  250  r.p.m. 

Calculate  as  much  of  this  design  as  time  permits.  If,  however, 
you  do  not  get  on  well  with  the  entire  design,  then  take  some 
particular  part  of  the  design,  say — the  estimation  of  the  magneto- 
motive force  and  stator  winding  and  carry  it  out  in  detail. 

While  there  is  still  time,  bring  together  the  leading  dimensions 
and  properties  in  a  concise  schedule.  Make  use  of  the  printed 
specifications  (to  be  had  on  request)  if  desired. 

PAPER  NUMBER  IV 

A  no-load  saturation  curve  is  given  in  Fig.  118.  This  applies 
to  a  three-phase  alternator  with  a  rated  output  of  850  kw.  at 


EXAMPLES  FOR  PRACTICE  IN  DESIGNING    229 

2880  volts  per  phase  and  94  r.p.m.  32  pole,  25-cycle  and  unity 
power-factor.  At  rated  load  the  reactance  voltage  is  945  volts 
per  phase,  and  the  resultant  maximum  armature  strength  is 
3900  ampere-turns  per  pole. 

Estimate  the  inherent  regulation  of  this  machine  for  rated  full- 
load  current  of  98.5  amperes  at  unity  power-factor.  Also  for 
this  same  current,  but  at  a  power-factor  of  0.8. 


Phase  Pressure 

^ 

(V 

„-•— 

.--—  • 

9>« 

^ 

-" 

? 

& 

1^*       850  K.V.A..32  Poles,  94  R.P.M. 
2880  Volts  per  Phase,  25  Cycles 

Reactance  Voltage  per  Phase  at  this  load(unity  P.F.J915  V 
Resultant  Armature  Strength                       3900  Maximum 
rms.  Current  per  Phase  at  Full  Load    and  Unity 
Power  Factor  =  98.5_Ampa. 
Turns  per  Phase  =  448 

/ 

/ 

/ 

/ 

/ 

1 

1 

/ 

/ 

/ 

/ 

2000     4000      6000     8000     10000  12000    14000    16000 
Ampere  Turns  per  Field  Spool 

FIG.  118. — No-load  Saturation  Curve  for  the  850-kva.  Three-phase  Generator 
Described  in  Paper  No.  IV. 

Draw  a  group  of  saturation  curves  (as  indicated  roughly  in 
Fig.  119,  for  98.5  amperes  and  for  power-factors  of  1.0,  0.8 
andO. 

Plot  excitation  regulation  curves  of  this  machine  for  normal 
pressure  of  2880  volts  per  phase,  and  for  power-factors  of  1.0, 
0.8  and  0. 

PAPER  NUMBER  V 

Answer  only  one  of  the  two  following  questions: 
(Do  not  necessarily  attempt  to  do  more  than  arrive  at  the 
approximate  general  outline  for  the  machine.     If,   afterwards, 
you  have  time,  work  out  and  tabulate  its  leading  properties. 


230       POLYPHASE  GENERATORS  AND  MOTORS 

But  the  chief  consideration  is  that  you  should  demonstrate  your 
ability  to  make  a  rough  estimate  of  the  most  probably  correct 
design.) 

1.  Design  a  4-pole,  Y-connected,  30-cycle,  three-phase,  squir- 
rel-cage induction  motor  for  a  primary  terminal  pressure  of  1000 
volts  (577  volts  per  phase)  and  for  a  rated  output  of  100  h.p. 


FIG.  119. — Rough  Indication  of  the  Saturation  Curves  Called  for  in  Paper 

No.  IV. 

2.  Design  a  50-cycle,  Y-connectea  poiypiiase  generator  for 
a  rated  output  of  1500  kva.  at  a  speed  of  375  r.p.m.  and  for  a  ter- 
minal pressure  of  5000  volts  (2880  volts  per  phase). 

PAPER  NUMBER  VI 

For  the  induction  motor  shown  in  Fig.  120: 

1.  Make  a  rough  estimate  of  a  reasonable  normal  output  to 
assign  to  the  motor. 

2.  Estimate  the  no-load  current. 

3.  Estimate  the  circle  ratio. 

4.  Estimate  the  breakdown  factor  at  the  output  you  have 
assigned  to  the  motor. 

5.  Estimate  the  losses  and  efficiency  at  the  output  you  have 
assigned  to  the  motor. 

6.  Estimate  the  temperature  rise  at  the  output  you  have 
assigned  to  the  motor. 

7.  Estimate  the  power-factor  at  various  loads. 


EXAMPLES  FOR  PRACTICE  IN  DESIGNING    231 

(NOTE. — If,  rightly  or  wrongly,  you  consider  that  some  essential  data 
have  not  been  included  in  Fig.  120,  do  not  lose  time  over  the  matter,  but 
make  some  rational  assumption — stating  on  your  paper  that  you  have  done 
so — and  then  proceed  with  the  work.) 


Rotor  Slot 


Cond. 
6.5  x  2.5mm 


THREE-PHASE  SQUIRREL-CAGE  INDUCTION  MOTOR. 


Number  of  poles 

Terminal  pressure 

(  .'.   Pressure  per  phase .  . 


Synchronous  speed  in  r.p.m. 
Stator  connections 


1000 
V* 


12 

1000  volts 

=   578  volts 

500 
Y 


Stator. 

Number  of  slots 180 

Dimensions  of  slot  (depth  Xwidth)   25  X 10 . 5  mm. 

Slot  opening 3  mm. 

Conductors  per  slot 6 

Dimensions  of  bare  conductor.  .  .  .    2.5  X6.5  mm. 
Number  of  end  rings 


Rotor. 
216 

21.5X8.0  mm. 
3  mm. 

4.5X16.0  mm. 
2 


Section  of  end  ring 20  X20  mm. 

Dimensions  in  centimeters  and  millimeters. 

FIG.  120.— Sketches  and  Data  of  Induction  Motor  of  Paper  No.  VI. 


PAPER  NUMBER  VII 

Design    the    following    three-phase,    squirrel-cage    induction 
motor: 

Rated  output  =  30  h.p.; 
Synchronous  speed  =  1000  r.p.m.; 
Periodicity  =  50  cycles  per  second; 
Pressure  between  terminals  =  500  volts; 
Y-connected  stator  winding. 

I .'.     Pressure  per  phase  =  — ^= = 288  volts. ) 


232       POLYPHASE  GENERATORS  AND  MOTORS 

Proportion  the  squirrel-cage  rotor  for  4  per  cent  slip  at  rated 
load.  Carry  the  design  as  far  as  time  permits,  but  devote  the 
last  half  hour  to  preparing  an  orderly  table  of  your  results. 

NOTE. — If,  rightly  or  wrongly,  you  conclude  that  some  essential  data  have 
not  been  included  in  the  above,  do  not  lose  time  over  the  matter,  but  make 
some  rational  assumption — stating  on  your  paper  that  you  have  done  so — 
and  then  proceed  with  the  work. 


PAPER  NUMBER  VIII 

For  the  induction  motor  design  of  which  data  is  given  in  Fig. 
121,  estimate  the  losses  at  rated  load.     Draw  its  circle  diagram. 

Stator  Slot 
|< 2.16 »| 


3,Ducts 
each  1.3  wide 


Dimensions  in  cm 


Dimensions  in  cm. 

Output  in  H.P 

Number  of  poles 

Connection  of  stator  windings 

Periodicity  in  cycles  per  second 

Volts  between  terminals 

Stator  Winding. 

Total  number  of  stator  conductors .  .  . 
Number  of  stator  conductors  per  slot . 


Dimensions  of  stator  conductor  (bare) .  . 


Mean  length  of  stator  turn 

Rot-ir  Windings. 

Total  number  of  rotor  conductors .... 
Number  of  rotor  conductors  per  slot.  . 
Number  of  phases  in  rotor  winding.  .  . 
Dimensions  of  rotor  conductor  (bare) . 
Mean  length  of  rotor  turn 


Rotor  Slot 
Rotor  Slot. 
.    220 
12 

.      Y 

50 

.5000 

.3564 

.    33,  each  consisting  of  two 

components 
.      2  in  parallel,  each  2.34  mm. 

diameter. 
,    159  cm. 

.    288 

2 

3 

.    16X9  mm. 
.    150  cm. 


FIG.  121. — Sketches  and  Data  of  the  220-H.P.  Induction   Motor   of  Paper 

No.  VIII. 

Plot  its  efficiency,  power-factor  and  output,  using  amperes  input 
as  abscissae,  from  no  load  up  to  the  breakdown  load.     Estimate 


EXAMPLES  FOR  PRACTICE  IN  DESIGNING    233 

the  starting  torque  and  the  current  input  to  the  motor  at  starting, 
when  half  the  normal  voltage  is  applied  to  the  terminals  of  the 
motor. 

NOTE. — If  you  are  of  opinion  that  sufficient  data  have  not  been  given  you 
to  enable  all  the  questions  to  be  answered,  do  not  hesitate  to  make  some 
reasonable  assumption  for  such  missing  data. 

PAPER  NUMBER  IX 

Design  a  300-h.p.,  three-phase,  40-cycle,  240-r.p.m.,  squirrel- 
cage  induction  motor  for  a  terminal  pressure  of  2000  volts.  Let 
the  stator  be  Y-connected.  Obtain  y,  the  ratio  of  the  no-load  to 
the  full-load  current,  a,  the  circle  ratio,  and  bdf.,  the 'break- 
down factor;  and  further  data  if  time  permits.  Employ  the 
last  half  hour  in  criticising  your  own  design,  and  in  stating  the 
changes  you  would  make  with  a  view  to  improving  it,  if  you  had 
.time. 

PAPER  NUMBER  X 

During  the  entire  six  hours  at  your  disposal,  design  a  three- 
phase  squirrel-cage  induction  motor,  to  the  following  specification: 

Normal  output  in  h.p 150 

Periodicity  in  cycles  per  second 50 

Synchronous  speed  (i.e.  speed  at  no  load),  in  r.p.m 250 

Terminal  pressure  in  volts 400 

Connection  of  phases Y 

Pressure  per  phase  in  volts .  231 

The  last  two  hours,  i.e.,  from  3  to  5  P.M.,  must  be  devoted 
to  tabulating  the  results  at  which  you  have  arrived,  and  to  the 
preparation  of  outline  sketches  with  principal  dimensions.  As 
to  the  electrical  design,  the  following  particulars  will  be  expected 
to  be  worked  out  or  estimated: 

1.  Ratio  of  no-load  to  full-load  current,  (y). 

2.  Circle  ratio,  (a). 

3.  Circle  diagram  to  scale. 

4.  Breakdown  factor  (bdf.). 

5.  Per  cent  slip  at  rated  load. 

6.  Losses  in  stator  winding,  at  rated  load. 


234       POLYPHASE  GENERATORS  AND  MOTORS 

7.  Losses  in  rotor  winding  (i.e.,  squirrel-cage  losses  at  rated 
load). 

8.  Core  losses. 

9.  Friction. 

10.  Efficiency  at  rated  load. 

11.  Power-factor  at  rated  load. 

12.  Estimate  of  thermometrically  determined  ultimate  tem- 
perature rise  at  rated  load. 

(NOTE. — The  students  are  permitted  to  bring  in  any  books  and  notes 
and  drawing  instruments  they  wish.  They  are  also  permitted  to  fill  out 
and  to  hand  in  as  a  portion  of  their  papers,  specification  forms  which  they 
have  prepared  in  advance  of  coming  to  the  examination.) 

PAPER  NUMBER  XI 

Any  notes  or  note-books  and  other  books  may  be  used,  but 
students  are  put  on  their  honor  not  to  discuss  any  part  of  their 
work  in  the  lunch  hour. 

A  three-phase,  squirrel-cage  induction  motor  complies  with 
the  following  general  specifications: 

Rated  load  in  h.p =  90 

Periodicity  in  cycles  per  second =40 

Speed  in  r.p.m.  at  synchronism =800 

Terminal  pressure =  750 

Connection  of  phases =  Y 

(1)  Given   the    following  data,  estimate  y,  the  ratio  of   no- 
load  to  full-load  current: 

Average  air-gap  density =3610  lines  per  sq.cm. 

Air-gap  depth =0.91  mm. 

Air-gap  mmf .  -f-  total  mmf =  0 . 80 

Total  number  of  conductor  on  stator ....  =  576 . 

(2)  Given  the  following  data,  estimate  a,  the  circle  ratio: 

Internal  diameter  of  stator  laminations =483  mm. 

Slot  pitch  of  stator  at  air-gap =21.1  mm. 

Slot  pitch  of  rotor  at  air-gap =24.8  mm. 

Gross  core  length;  \g =430  mm. 


EXAMPLES  FOR  PRACTICE  IN  DESIGNING    235 

(3)  Draw    the    circle    diagram.       Determine    the    primary 
current : 

(a)  At  point  of  maximum  power-factor. 
(6)  breakdown  load. 

(4)  Plot  curves  between: 

(a)  H.p.  and  YJ,  the  efficiency. 
(6)  H.p.  and  G,  the  power-factor. 

(c)  H.p.  and  slip. 

(d)  H.p.  and  amperes  input. 

Given  the  following  data: 

Section  of  stator  conductor  =0.167  sq.cm. 

Dimensions  of  rotor  conductor  =1.27  cm.X0.76  cm. 

(One  rotor  conductor  per  slot) 

Length  of  each  rotor  conductor  =49.5  cm. 

Diameter  of  end  rings  (external)  =45  em. 

Number  of  end  rings  at  each  end  =  2,  each  2.54  cm.  X 0.63  cm. 

Total  constant  losses  =2380  watts. 

(5)  Calculate  the   starting   current   and   torque  when   com- 
pensators supplying  33,  40  and  60  per  cent  of  the  terminal  pres- 
sure are  used. 

(6)  Calculate  new  end  rings  so  that  with  a  2  :  1  compensator 
the  starting  torque  shall  be  equal  to  one-half  torque  at  rated  load. 

(7)  Designating  the  first  motor  as  A   and  the  second  one 
(i.e.,  the  modification  obtained  from  Question  6)  as  B,  tabulate 
the  component  losses  at  full  load  in  two  parallel  (vertical  columns) . 

Estimate    the    watts    total  loss   per    ton    weight    of   motor 
(exclusive  of  side-rails  and  pul'ey). 

(8)  If  the  squirrel-cage  of  the  first  motor  A  is  replaced  by 
a  three-phase   winding   having   the   same   equivalent   losses   at 
full  load,    calculate   the   resistance   per  phase   which   would   be 
required,  external  to  the  slip  rings,  in  order  to  limit  the  starting 
current  to  70  amperes.     What  would  be  the  starting  torque, 
expressed  as  percentage  of  full-load  torque,  with  this  external 
resistance  inserted? 


236       POLYPHASE  GENERATORS  AND  MOTORS 

PAPER  NUMBER  XII 

For  the  three-phase  induction  motor  of  which  data  is  given 
below,  make  calculations  enabling  you  to  plot  curves  with 
amperes  input  per  phase  as  abscissae  and  power-factor,  efficiency 
and  output  in  h.p.  as  ordinates. 

Rated  load =60  h.p. 

Periodicity  in  cycles  per  second .    =  50 

Speed  at  synchronism =  600  r.p.m. 

Terminal  pressure =  550  volts 

Connection  of  phases =  Y 

Average  air-gap  density =3900  lines  per  sq.cm. 

Air-gap  depth =0.  9  mm. 

Total  mmf.  -f-  air-gap  mmf =  1.  2 

Total  no.  of  conductors  on  stator   =  720 

Circle  ratio =0.061 

I2R  losses  at  rated  load =2840 

Constant  losses =  2050 

PAPER  NUMBER  XIII 

The  data  given  below  are  the  leading  dimensions  of  the  stator 
and  rotor  of  a  24-pole  three-phase  induction  motor  with  a  Y-con- 
nected  winding  suitable  for  a  25-cycle  circuit.  Ascertain  approx- 
imately by  calculation  the  suitable  terminal  voltage  for  this 
induction  motor  and  give  your  opinion  of  the  suitable  rated  out- 
put. Proceed,  as  far  as  time  permits,  with  the  calculation  of 
the  circle  ratio  and  of  the  no-load  current,  and  construct  the 
circle  diagram.  Abbreviate  the  calculations  as  much  as  possible 
by  reasonable  assumptions  for  the  less  important  steps,  thus 
obtaining  more  time  for  the  steps  where  assumptions  can  les§ 
safely  be  made. 

DIMENSIONS  IN  MM. 
Stator. 

External  diameter 2800 

Internal  diameter 2440 

Gross  core  length 385 

Net  core  length 299 

Winding 3-phase,  Y-connected 

Number  of  slots 216 

Depth  X  width  of  slot 51X21 

Conductors  per  slot 12 

Section  of  conductors,  sq.cm 0.29 


EXAMPLES  FOR  PRACTICE  IN  DESIGNING    237 

Rotor. 

External  diameter 2434.5 

Internal  diameter 2144 

Winding 3-phase,  Y-connected 

Number  of  slots 504 

Depth  X  width  of  slot 35X9.5 

Conductors,  per  slot 2 

Section  of  conductor,  sq.cm 0.811 

[This  paper  to  be  brought  in  for  the  afternoon  examination  (see  Paper 
No.  XIV),  as  certain  data  in  it  will  be  required  for  the  afternoon  examina- 
tion.] 

PAPER  NUMBER  XIV 

The  24-pole  stator  which  you  employed  this  morning  (see 
Paper  No.  XIII),  for  an  induction  motor,  will,  if  supplied  with 
a  suitable  internal  revolving  field  with  24  poles,  make  an  excellent 
three-phase,  25-cycle,  Y-connected  alternator.  What  would 
be  an  appropriate  value  for  the  rated  output  of  this  alternator? 
Without  taking  the  time  to  calculate  it,  draw  a  reasonable  no- 
load  saturation  curve  for  this  machine.  From  this  curve  and  from 
the  data  of  the  machine  and  your  assumption  as  to  the  appro- 
priate rating,  calculate  and  plot  a  saturation  curve  for  the  rated 
current  when  the  power-factor  of  the  external  circuit  is  0.80. 

PAPER  NUMBER  XV 

1.  Of  two  50-cycle,  100-h.p.,  500-volt,  three-phase  induction 
motors,  one  has  4  poles  and  the  other  has  12  poles. 

(a)  Which  will  have  the  higher  power-factor? 
(6)  "  "  "  efficiency? 

(c)  "  li  "  current  at  no  load? 

(d)  "  "  "  breakdown  factor? 

Of  two  750  r.p.m.,  100-h.p.,  500-volt,  three-phase  induction 
motors,  one  is  designed  for  50  cycles,  and  the  other  for  25  cycles, 

(e)  Which  will  have  the  higher  power-factor? 

(/)  "  "  "  current  at  no  load? 

(0)  li  "  "  breakdown  factor? 

2.  Describe  how  to  estimate  the  temperature  rise  of  an  induc- 
tion motor. 

3.  Describe  how  to  estimate  the  magnitude  and  phase  of  the 
starting  current  of  a  squirrel-cage  induction  motor. 


238       POLYPHASE  GENERATORS  AND  MOTORS 


PAPER  NUMBER  XVI 

Answer  one  of  the  following  two  questions. 
Question  I.  For  the  three-phase  squirrel-cage,  induction  motor 
of  which  data  are  given  in  Fig.  122: 


Rotor  Slot 


h-i3,H 

mm 


Terminal  pressure 750 

Method  of  connection Y 

Pressure  per  phase 432 

Speed  in  r.p.m SOO 

Full  load  primary  current  input  per  phase.  .  . '.  59 

Number  of  primary  conductors  per  slot 8 

Periodicity  in  cycles  per  second 40 


Length  in  cm. 

Density  in 
Lines  per  sq.cm. 

Ampere-turns 
per  cm.  of 
Length. 

Total  Ampere- 
turns. 

Stator  teeth  

Stator  core  .... 

Rotor  teeth  

Rotor  core  
Air-gap 

Total  mmf.  per  pole 

FIG.  122. — Sketches  and  Data  of  the  Three-phase  Induction  Motor  of  Question 
I  in  Paper  No.  XVI. 

(a)  Work  out  the  magnetic  circuit  calculations  required  to 
fill  in  the  table  indicated,  and  enter  up  the  results  in  a  similar 
table. 

(b)  Obtain  y  and  a  for  the  motor. 


EXAMPLES  FOR   PRACTICE  IN  DESIGNING    239 

Question  II.  For  a  certain  three-phase,  slip  ring,  induction 
motor  the  following  data  apply : 

Periodicity  in  cycles  per  sec 50 

Speed  at  no  load  in  r.p.m 500 

Y  (ratio  of  no-load  to  full-load  current) 0 . 36 

a  (circle  ratio) 0.0742 

Rated  output  in  h.p 300 

Terminal  pressure 700 

Pressure  per  phase 405 

Connection  of  phases Y 

Stator  resistance  per  phase  (ohm) 0 . 030 

Rotor  resistance  per  phase  (ohm) 0 . 022 

Total  core  loss  (watt) 4080 

Friction  and  windage  loss  (watt) 2000 

Ratio  number  of  stator  to  number  of  rotor  conductors ...  1 . 28 

(a)  Draw  the  circle  diagram. 

(6)  Plot  curves  of  efficiency,  power-factor,  output  in  horse- 
power and  slip,  all  as  a  function  of  the  current  input  per  phase. 

PAPER  NUMBER  XVII 

1.  For  an  armature  having  an  air-gap  diameter,  Z)  =  65  cms. 
and  a  gross  core  length  \g  =  35  cms. 

What  would  be  a  suitable  rating  for  a  500-volt,  500-r.p.m. 
machine  of  these  data: 

1st.  As  a  25-cycle  induction  motor. 

2d.  As  a  25-cycle  alternator. 

Select  one  of  these  cases  and  work  out  the  general  lines  of  the 
design  as  far  as  time  permits. 

2.  In  Fig.  123  are  given  data  of  the  design  of  an  alternator 
for  the  following  rating: 

2500  kva.,  3-phase,  25-cycle,  75  r.p.m.,  6500-volts,  Y-con- 
nected.  The  field  excitations  for  normal  voltage  of  2200  volts  and 
for  1.2  times  normal  voltage  (2640  volts),  are  given.  From 
these  values  the  no-load  saturation  curve  may  be  drawn. 

Estimate  (showing  all  the  necessary  steps  in  the  calculations). 

(a)  The  field  ampere-turns  required  for  full  terminal  voltage 
at  full-load  kva.  at  power-factors  1.0  and  0.8;  and  the  pressure 
regulation  for  both  these  cases. 


240       POLYPHASE  GENERATORS  AND  MOTORS 

(b)  The    short    circuit    current    for    normal  speed  and    with 
no-load  field  excitation  for  normal  voltage. 

(c)  The  losses  and  efficiency  at  rated  full  load  and  J  load 
at  0.8  power-factor.     Also  the  armature  heating. 


Scale  1-20 


Mh          25°° 


-  65°°  V<>lt 

Y  Connected  A.C.  Generator 


Data 

No  of  Conductors  per  Slot  9 

True  Cross  Section  of  1  Conductor      1.29  sq.cm. 

Field  Spool  Winding 

Turns  per  Spool  42.5 

Mean  Length  of  1  Turn  1620mm 

Cross  Section  of  Conductor  1.78  sq.  cms 

Saturation-  Ampere  Turns  at  6500  Volts  8000 
,«  ..      ..  7800      ...    13000 

Air  Gap  Ampere  Turns  at  6500  Volts       goOO 


FIG.  123. — Data  of  the  Design  of  the  2500-kva.  Alternator  of  Question  2 
of  Paper  No.  XVII. 

3.  In  Fig.  121  are  given  data  of  the  design  of  an  induction 
motor  for  the  following  rating : 

40-h.p.,  600-r.p.m.,  50-cycles,  500-volt  A-connected,  three- 
phase  induction  motor. 


Scale  1-10 


Stator  Slot  opening  4mm 
Rotor  Slot  opening  l.Smn 


40  H.P.  600  R.P.M.  50f\J500  Volt,  3  Phase  Induction  Motor 

Stator  Winding  Rotor  Winding 

Connection  Winding  A  Squirrel  Cage  Type 

No.of  Conductors  pet-  Slofr  18  Total  No. of  Bars  39 

Cross  Section  of  1  Conductor  0.0685  sq.cms 

*   Friction  and  Windage  Losses  300 


All  Dimensions 

Total  No.of  Bars  39  in  Millimeters 

Cross  Section  of  Bar        1.0  sq.cm 
Cross  Section  of  End  Ring    3.0  sq.cm 


FIG.  124.— Data  of  the   Design   of  the  40-H.P.   Motor  of   Question  3  of 

Paper  No.  XVII. 

Estimate  (showing  all  the  necessary  steps  in  the  calculations). 

(a)  Circle  ratio  ( a) ; 

(b)  No-load  current  in  per  cent  of  full-load  current; 

(c)  Breakdown  factor  ; 

(d)  Maximum  power-factor. 


EXAMPLES  FOR  PRACTICE  IN  DESIGNING     241 

(e)  Losses  and  efficiencies  and  heating  at  \  load  and  at  rated 
full  load. 

(/)  Slip  at  full  load. 

PAPER  NUMBER  XVIII 

(You  may  use  notes,  curves  or  any  other  aids.) 
1.  The  leading  particulars  of  a  certain  induction  motor  are 
given  in  Fig.  125. 


80.  H.P.,  600  r.p.m.,  50   CYCLES,  500  VOLTS,  A-CONNECTED  SQUIRREL-CAGE  INDUCTION 

MOTOR 

Data: 

Number  of  stator  slots 90 

f  Depth 36.0 

Slot  dimensions  <   Width 11.0 

[  Opening 6.0 

Number  of  rotor  slots Ill 

f  Depth 21.5 

Slot  dimensions  {   Width 6.5 

[  Opening 1.5 

WINDINGS: 
Stator 

Conductors  per  slot 12 

Cross-section  of  conductor 0. 138  sq.cm. 

Rotor: 

Bars  per  slot 1 

Cross-section  of  bar 1.0  sq.cm. 

Cross-section  of  end  ring 3.6  sq.cm. 

All  dimensions  in  millimeters. 

FIG.  125. — Sketches  and  Data  of  the  80-H.P.  Induction  Motor  of  Question  1 
of  Paper  No.  XVIII. 


(a)  Estimate  y  ti\e  ratio  of  the  no-load  current  to  the  current 
at  normal  rating. 

(6)  Estimate  a  the  circle  ratio,  and  also  estimate  the  maximum 
power-factor. 

(c)  Estimate  bdf.,  the  breakdown  factor. 


242       POLYPHASE  GENERATORS  AND  MOTORS 

(d)  Estimate  the  copper  losses  at  normal  rating. 

(e)  Estimate  the  core  losses  at  normal  rating. 

(/)  Estimate  the  watts  per  square  decimeter  of  equivalent  gap 
surface. 

(g)  Estimate  the  efficiency  at  normal  rating. 

(h)  Estimate  the  power-factor  at  normal  rating. 

(i)  Estimate  the  T.W.C.  (the  total  works  cost). 

2.  (a)  Deduce  the  leading  proportions  for  a  three-phase 
alternator  for  a  normal  rating  of  1500  kw.,  1000  r.p.m.,  50  cycles, 
6  poles,  11  000  volts.  Do  not  go  into  detail,  but  go  far  enough  to 
give  an  opinion  as  to  the  suitable  values  for  D,  \g,  number  of 
slots,  conductors  per  slot,  and  field  excitation  at  no  load.  State 
briefly  your  reasons  for  choosing  each  of  these  values. 

(6)  What,  in  a  general  way,  should  be  the  changes  in  the 
general  order  of  magnitude  of  these  quantities  for  a  design  for 
the  same  rated  output  and  voltage  at 

(1)     50  cycles,  250  r.p.m.,  24  poles, 

and  the  changes  necessary  in  this  second  design  in  order  to  obtain 
a  design  for 

(2)     25  cycles,  250  r.p.m.,  12  poles. 

PAPER  NUMBER  XIX 

Work  up  a  rough  outline  for  a  design  for  a  25-cycle,  three- 
phase  generator  to  supply  at  a  pressure  of  10  000  volts  (Y-con- 
nected  with  5770  volts  per  phase).  The  generator  is  to  have  a 
rated  capacity  of  3000  kva.  at  a  power-factor  of  0.90,  and  is  to 
be  run  at  a  speed  of  125  r.p.m.  Design  the  machine  to  give  good 
regulation  of  the  pressure,  and  work  out  the  inherent  regulation 
at  various  power-factors.  Work  out  any  other  data  for  which 
you  find  time,  and  devote  the  last  half  hour  to  an  orderly  tabu- 
lation of  your  results. 

i 
PAPER  NUMBER  XX 

Work  up  a  rough  outline  for  a  design  fc/r  a  25-cycle,  8-pole, 
three-phase,  squirrel-cage  induction  motor  for  a  rated  output 
of  80  h.p. 

The  terminal  pressure  is  600  volte,  i.e.,  346  volts  per  phase, 


EXAMPLES  FOR  PRACTICE  IN  DESIGNING     243 

and  the  s  tat  or  windings  should  be  Y-connected.  The  load  will 
not  be  thrown  on  the  motor  until  it  is  up  to  speed. 

During  the  last  half  hour  prepare  an  orderly  tabulation  of 
the  leading  results  which  you  have  found  time  to  work  out. 

PAPER  NUMBER  XXI 

Commence  the  design  of  a  three-phase,  100-h.p.,  squirrel- 
cage  induction  motor  with  a  Y-connected  stator  for  a  synchro- 
nous speed  of  375  r.p.m.  when  operated  from  a  25-cycle  circuit 
with  a  line  pressure  of  500  volts.  (The  pressure  per  stator  wind- 

500 
ing  is  consequently  — ^==288  volts.)     Try  and  carry  the  design 

as  far  as  determining  upon  the  gap  diameter  and  the  gross  core 
length,  the  number  of  stator  slots,  the  number  of  stator  conductors 
per  slot,  the  flux  per  pole  and  the  external  diameter  of  the  stator 
laminations  and  the  internal  diameter  of  the  rotor  laminations. 
Then  tabulate  these  data  in  an  orderly  manner  before  proceeding 
further.  Then,  if  time  permits,  make  the  magnetic  circuit 
calculations  and  estimate  the  magnetizing  current, 


PAPER  NUMBER  XXII 

(Answer  one  of  the  following  two  questions.) 

1.  Which  would  be  the  least  desirable,  as  regards  interfering 
with  the  pressure  on  a  50-cycle  net  work,  low-speed  or  high- 
speed induction  motors?  Why? 

Of  two  1000-volt,  100-h.p.,  750-r.p.m.,  three-phase  induction 
motors,  which  would  have  the  highest  capacity  for  temporarily 
carrying  heavy  overloads,  a  25-cycle  or  a  50-cycle  design?  Wliich 
would  have  the  highest  power-factor?  Which  the  lowest  current 
when  running  unloaded? 

Of  two  1000-volt,  100-h.p.,  three-phase  induction  motors 
for  25  cycles,  one  is  for  a  synchronous  speed  of  750  r.p.in.  and  the 
other  is  for  a  synchronous  speed  of  150  r.p.m.  Assuming  rational 
design  in  both  cases,  which  has  the  higher  power-factor?  Which 
the  lower  current  when  running  light?  Which  the  higher 
breakdown  factor?  Which  the  higher  efficiency?  Three  100- 


244       POLYPHASE  GENERATORS  AND  MOTORS 

h.p.,  1000- volt  designs  have  been  discussed  above.     Their  speeds 
and  periodicities  may  be  tabulated  as  follows: 

DpdirnatJrm  Synchronous  speed  Periodicity  in  cycles 

inr.p.m.  per  sec. 

A  750  25 

B  750  50 

C  150  25 

Assume  that  these  are  provided  with  low-resistance  squirrel- 
cage  windings.  Make  rough  estimations  of  the  watts  total  loss 
per  ton  of  total  weight  of  motor  for  each  case. 

2.  A  three-phase,  Y-connected,  squirrel-cage  induction  motor 
has  the  following  constants: 

External  diameter  stator  core 1150  mm. 

Air-gap  diameter  (D) 752  mm. 

Internal  diameter  rotor  core 324  mm. 

Diameter  at  bottom  of  stator  slots ...  841  mm. 

Diameter  at  bottom  of  rotor  slots 710  mm. 

No.  of  stator  slots 72 

No.  of  rotor  slots 89 

Conductors  per  stator  slot 5 

Conductors  per  rotor  slot    1 

Width  of  stator  slot 19  mm. 

Width  of  rotor  slot 12  mm. 

Space  factor  stator  slot 0.51 

Space  factor  rotor  slot 0.80 

Gross  core  length  (Xgr) 190  mm. 

Net  core  length  (Xn) 148  mm. 

Polar  pitch  (T) 394  mm. 

Depth  of  air-gap  (A) 1.5  mm. 

Peripheral  speed 38.6  mps. 

Output  coefficient  (£) 2.37 

The  pressure  per  phase  is  1155  volts,  the  pressure  between 
terminals  being  2000  volts. 

What  is  the  rated  output  of  the  motor  in  h.p.? 
What  is  the  speed  in  r.p.m.? 
What  is  the  periodicity  in  cycles  per  second? 
What  is  the  flux  per  pole  in  megalines? 
What  is  the  stator  PR  loss? 


EXAMPLES  FOR  PRACTICE  IN  DESIGNING     245 

What  is  the  stator  core  loss? 

What  cross-section  must  be  given  to  the  copper  end  rings 
in  order  that  the  slip  at  rated  load  shall  be  2  per  cent? 

Estimate  the  efficiency  at  J,  J  and  full  load. 

Estimate  a  the  circle  ratio. 

Estimate  Y  the  ratio  of  the  magnetizing  current  to  the  current 
at  rated  load. 

What  is  the  power-factor  at  }-,  J  and  full  load? 

What  is  the  breakdown  factor? 

Estimate  the  probable  temperature  rise  after  continuous 
running  at  rated  load. 

Remember  that  some  of  these  questions  are  most  readily 
solved  by  constructing  a  circle  diagram  and  combining  the 
dimensions  scaled  off  from  it,  with  slide-rule  calculations. 


APPENDIX  I 


A  BIBLIOGRAPHY  OF  I.  E.  E.  AND  A.  I.  E.  E.  PAPERS  ON 
THE  SUBJECT  OF  POLYPHASE  GENERATORS. 


1891. 

M.  I.  PUPIN. — On  Polyphase  Generators.     (Trans.  Am.  Inst.  Elec. 
Engrs.,  Vol.  8,  p.  562). 

1893. 

GEORGE    FORBES. — The    Electrical    Transmission    of    Power    from 
Niagara  Falls.     (Jour.  Inst.  Elec.  Engrs.,  Vol.  22,  p.  484.) 

1898. 

A.  F.  McKissiCK. — Some  Tests  with  an  Induction  Generator.    (Trans. 
Am.  Inst.  Elec.  Engrs.,  Vol.  15,  p.  409.) 

1899. 

M.  R.  GARDNER  and  R.  P.  HOWGRAVE-GRAHAM. — The  Synchronizing 
of  Alternators.     (Jour.  Inst.  Elec.  Engrs.,  Vol.  28,  p.  658.) 

1900. 

B.  A.  BEHREND. — On  the  Mechanical  Forces  in  Dj^namos  Caused 
by  Magnetic  Attraction.     (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  17,  p.  617.) 

1901. 

W.  L.  R.  EMMET. — Parallel  Operation  of  Engine-Driven  Alternators. 
(Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  18,  p.  745.) 

ERNST  J.   BERG. — Parallel  Running  of  Alternators.     (Trans.   Am. 
Inst.  Elec.  Engrs.,  Vol.  18,  p.  753.) 

247 


248       POLYPHASE  GENERATORS  AND  MOTORS 

P.  0.  KEILHOLTZ. — Angular  Variation  in  Steam  Engines.  (Trans. 
Am.  Inst.  Elec.  Engrs.,  Vol.  18,  p.  703.) 

CHAS.  P.  STEINMETZ. — Speed  Regulation  of  Prime  Movers  and 
Parallel  Operation  of  Alternators.  (Trans.  Am.  Inst.  Elec.  Engrs. 
Vol.  18,  p.  741.) 

WALTER  I.  SLIGHTER. — Angular  Velocity  in  Steam  Engines  in  Rela- 
tion to  Paralleling  of  Alternators.  (Trans.  Am.  Inst.  Elec.  Engrs., 
Vol.  18,  p.  759.) 

1902. 

C.  0.  MAILLOUX. — An  Experiment  with  Single-Phase  Alternators  on 
Polyphase  Circuits.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  19,  p.  851.) 

Louis  A.  HERDT. — The  Determination  of  Alternator  Characteristics. 
(Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  19,  p.  1093.) 

C.  E.  SKINNER. — Energy  Loss  in  Commercial  Insulating  Materials 
when  Subjected  to  High-Potential  Stress.  (Trans.  Am.  Inst.  Elec. 
Engrs.,  Vol.  19,  p.  1047.) 

1903. 

C.  A.  ADAMS. — A  Study  of  the  Heyland  Machine  as  Motor  and  Gen- 
erator. (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  21,  p.  519.) 

W.  L.  WATERS. — Commercial  Alternator  Design.  (Trans.  Am.  Inst. 
Elec.  Engrs.,  Vol.  22,  p.  39.) 

A.  S.   GARFIELD. — The  Compounding  of  Self-Excited  Alternating- 
Current  Generators  for  Variation  in  Load  and  Power  Factor.     (Trans. 
Am.  Inst.  Elec.  Engrs.,  Vol.  21,  p.  569.) 

B.  A.  BEHREND. — The  Experimental  Basis  for  the  Theory  of  the 
Regulation  of  Alternators.     (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  21, 
p.  497.) 

1904. 

A.  F.  T.  ATCHISON. — Some  Properties  of  Alternators  Under  Various 
Conditions  of  Load.     (Jour.  Inst.  Elec.  Engrs.,  Vol.  33,  p.  1062.) 

H.  W.  TAYLOR. — Armature  Reaction  in  Alternators.  (Jour.  Inst. 
Elec.  Engrs.,  Vol.  33,  p.  1144.) 

MILES  WALKER. — Compensated  Alternate-Current  Generators. 
(Jour.  Inst.  Elec.  Engrs.,  Vol.  34,  p.  402.) 

J.  B.  HENDERSON  and  J.  S.  NICHOLSON. — Armature  Reaction  in 
Alternators.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  34,  p.  465.) 

DAVID  B.  RUSHMORE. — The  Mechanical  Construction  of  Revolving 
Field  Alternators.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol  23,  p.  253.) 

B.  G.  LAMME.— Data  and  Tests  on  a  10  000  Cycle-per-Second  Alter- 
nator.    (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  23,  p.  417.) 


APPENDIX  249 

H.  H.  BARNES,  Jr. — Notes  on  Fly-Wheels.  (Trans.  Am.  Inst.  Elec. 
Engrs.,  Vol.  23,  p.  353.) 

H.  M.  HOBART  and  FRANKLIN  PUNGA. — A  Contribution  to  the 
Theory  of  the  Regulation  of  Alternators.  (Trans.  Am.  Inst.  Elec.  Engrs., 
Vol.  23,  p.  291.) 

1905. 

WILLIAM  STANLEY  and  G.  FACCIOLI. — Alternate-Current  Machinery, 
with  Especial  Reference  to  Induction  Alternators.  (Trans.  Am.  Inst. 
Elec.  Engrs.,  Vol.  24,  p.^51.) 

A.  B.  FIELD. — Eddy  Currents  in  Large,  Slot-wound  Conductors. 
(Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  24,  p.  761.) 

W.  J.  A.  LONDON. — Mechanical  Construction  of  Steam-Turbines  and 
Turbo-Generators.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  35,  p.  163.) 

1906. 

J.  EPSTEIN. — Testing  Electrical  Machinery  and  Materials.  (Jour. 
Inst.  Elec.  Engrs.,  Vol.  38,  p.  28.) 

A.  G.  ELLIS. — Steam  Turbine  Generators.     (Jour.  Inst.  Elec.  Engrs., 
Vol.  37,  p.  305.) 

SEBASTIAN  SENSTIUS. — Heat  Tests  on  Alternators.  (Trans.  Am. 
Inst.  Elec.  Engrs.,  Vol.  25,  p.  311.) 

MORGAN  BROOKS  and  M.  K.  AKERS. — The  Self-Synchronizing  of 
Alternators.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  25,  p.  453.) 

E.  F.  ALEXANDERSON. — A  Self-Exciting  Alternator.  (Trans.  Am. 
Inst.  Elec.  Engrs.,  Vol.  25,  p.  61.) 

1907. 

B.  A.  BEHREND. — Introduction  to  Discussion  on  the  Practicability  of 
Large  Generators  Wound  for  22000    Volts.     (Trans.  Am.  Inst.  Elec. 
Engrs.,  Vol.  26,  p.  351.) 

ROBERT  POHL. — Development  of  Turbo-Generators.  (Jour.  Inst. 
Elec.  Engrs.,  Vol.  40,  p.  239.) 

G.  W.  WORRALL. — Magnetic  Oscillations  in  Alternators.  (Jour.  Inst. 
Elec.  Engrs.,  Vol.  39,  p.  208.)  [This  paper  is  supplemented  by  another 
paper  contributed  by  Mr.  Worrall  in  1908.] 

1908. 

M.  KLOSS. — Selection  of  Turbo- Alternators.  (Jour.  Inst.  Elec. 
Engrs.,  Vol.  42,  p.  156.) 

S.  P.  SMITH. — Testing  of  Alternators.  (Jour.  Inst.  Elec.  Engrs., 
Vol.  42,  p.  190.) 


250       POLYPHASE  GENERATORS  AND  MOTORS 

G.  STONE Y  and  A.  H.  LAW. — High-Speed  Electrical  Machinery. 
(Jour.  Inst.  Elec.  Engrs.,  Vol.  41,  p.  286.) 

R.  K.  MORCOM  and  D.  K.  MORRIS. — Testing  Electrical  Generators. 
(Jour.  Inst.  Elec.  Engrs.,  Vol.  41,  p.  137.) 

G.  W.  WORRALL. — Magnetic  Oscillations  in  Alternators.  (Jour.  Inst. 
Elec.  Engrs.,  Vol.  40,  p.  413.)  [This  paper  is  a  continuation  of  Mr. 
Worrall's  1907  paper.  | 

JENS  BACHE-WIIG. — Application  of  Fractional  Pitch  Windings  to 
Alternating-Current  Generators.  (Trans.  Am.  Inst.  Elec.  Engrs., 
Vol.  27,  p.  1077.)  • 

CARL  J.  FECHHEIMER. — The  Relative  Proportions  of  Copper  and 
Iron  in  Alternators.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  27,  p.  1429.) 


1909. 

S.  P.  SMITH. — The  Testing  of  Alternators.  (Jour.  Inst.  Elec.  Engrs., 
Vol.  42,  p.  190.) 

J.  D.  COALES. — Testing  Alternators.  (Jour.  Inst.  Elec.  Engrs., 
Vol.  42,  p.  412.) 

E.  ROSENBERG. — Parallel  Operation  of  Alternators.  (Jour.  Inst. 
Elec.  Engrs.,  Vol.  42,  p.  524.) 

E.  F.  W.  ALEXANDERSON. — Alternator  for  One  Hundred  Thousand 
Cycles.  (Trans.  Am.  Inst.  Elec.  Erigrs.,  Vol.  28,  p.  399.) 

CARL  J.  FECHHEIMER. — Comparative  Costs  of  25-Cycle  and  60-Cycle 
Alternators.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  28,  p.  975.) 

C.  A.  ADAMS. — Electromotive  Force  Wave-Shape  in  Alternators. 
(Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  28,  p.  1053.) 


1910. 

MILES  WALKER. — Short-Circuiting  of  Large  Electric  Generators. 
(Jour.  Inst.  Elec.  Engrs.,  Vol.  45,  p.  295.) 

MILES  WALKER. — Design  of  Turbo  Field  Magnets  for  Alternate- 
Current  Generators.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  45,  p.  319.) 

GEO.  I.  RHODES. — Parallel  Operation  of  Three-Phase  Generators 
with  their  Neutrals  Interconnected.  (Trans.  Am.  Inst.  Elec.  Engrs., 
Vol.  29,  p.  765.) 

H.  G.  STOTT  and  R.  J.  S.  PIGOTT—  Tests  of  a  15  000-kw.  Steam- 
Engine-Turbine  Unit.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  29,  p.  183.) 

E.  D.  DICKINSON  and  L.  T.  ROBINSON. — Testing  Steam  Turbines 
and  Steam  Turbo-Generators.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  29, 
p.  1679.) 


APPENDIX  251 

1911. 

J.  R.  BARR. — Parallel  Working  of  Alternators.  (Jour.  Inst.  Elec. 
Engrs.,  Vol.  47,  p.  276.) 

A.  P.  M.  FLEMING  and  R.  JOHNSON. — Chemical  Action  in  the  Wind- 
ings of  High-Voltage  Machines.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  47, 
p.  530.) 

S.'  P.  SMITH.  —  Non-Salient-Pole  Turbo-Alternators.  (Jour.  Inst. 
Elec.  Engrs.,  Vol.  47,  p.  562.) 

W.  W.  FIRTH. — Measurement  of  Relative  Angular  Displacement  in 
Synchronous  Machines.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  46,  p.  728.) 

R.  F.  SCHUCHARDT  and  E.  0.  SCHWEITZER. — The  Use  of  Power- 
Limiting  Reactances  with  Large  Turbo-Alternators.  (Trans.  Am.  Inst. 
Elec.  Engrs.,  Vol.  30.) 

1912. 

H.  D.  SYMONS  and  MILES  WALKER. — The  Heat  Paths  in  Electrical 
Machinery.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  48,  p.  674.) 

W.  A.  DURGIN  and  R.  H.  WHITEHEAD. — The  Transient  Reactions  of 
Alternators.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  31.) 

A.  B.  FIELD. — Operating  Characteristics  of  Large  Turbo-Generators. 
(Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  31.) 

H.  M.  HOBART  and  E.  KNOWLTON. — The  Squirrel-Cage  Induction 
Generator.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  31.) 

E.  M.  OLIN. — Determination  of  Power  Efficiency  of  Rotating  Elec- 
tric Machines.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  31.) 

D.  W.  MEAD. — The  Runaway  Speed  of  Water-Wheels  and  its  Effect 
on  Connected  Rotary  Machinery.  (Trans.  Am.  Inst.  Elec.  Engrs., 
Vol.  31.) 

D.  B.  RUSHMORE. — Excitation  of  Alternating-Current  Generators. 
(Trans.  Am.  Inst,  Elec,  Engrs.,  Vol.  31.) 


APPENDIX  II 


A  BIBLIOGRAPHY  OF  I.  E.  E.  AND  A.  I.  E.  E.  PAPERS  ON 
THE  SUBJECT  OF  POLYPHASE  MOTORS 


1888. 

NIKOLA  TESLA. — A  New  System  of  Alternate-Current  Motors  and 
Transformers.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  5,  p.  308.) 

1893. 

ALBION  T.  SNELL.— The  Distribution  of  Power  by  Alternate-Current 
Motors.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  22,  p.  280.) 

1894. 

Louis  BELL. — Practical  Properties  of  Polyphase  Apparatus.  (Trans. 
Am.  Inst.  Elec.  Engrs.,  Vol.  11,  p.  3.) 

Louis  BELL. — Some  Facts  about  Polyphase  Motors.  (Trans.  Am. 
Inst.  Elec.  Engrs.,  Vol.  11,  p.  559.) 

Louis  DUNCAN,  J.  H.  BROWN,  W.  P.  ANDERSON,  and  S.  Q.  HAYES. — 
Experiments  on  Two-Phase  Motors.  (Trans.  Am.  Inst.  Elec.  Engrs., 
Vol.  11,  p.  617.) 

SAMUEL  REBER. — Theory  of  Two-  and  Three-Phase  Motors.  (Trans. 
Am.  Inst.  Elec.  Engrs.,  Vol  11,  p.  731.)  . 

CHAS.  P.  STEINMETZ. — Theory  of  the  Synchronous  Motor.  (Trans. 
Am.  Inst.  Elec.  Engrs.,  Vol.  11,  p.  763.) 

LUDWIG  GUTMANN. — On  the  Production  of  Rotary  Magnetic  Fields 
by  a  Single  Alternating  Current.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  11, 
p.  832.) 

1897. 

CHAS.  P.  STEINMETZ. — The  Alternating-Current  Induction  Motor. 
(Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  14,  p.  185.) 

252 


APPENDIX  253 

1899. 

ERNEST  WILSON. — The  Induction  Motor.     (Jour.  Inst.  Elec.  Engrs., 
Vol.  28,  p.  321.) 

1900. 

A.   C.   EBORALL. — Alternating  Current   Induction   Motors.     (Jour. 
Inst.  Elec.  Engrs.,  Vol.  29,  p.  799.) 


1901. 

CHAS.  F.  SCOTT. — The  Induction  Motor  and  the  Rotary  Converter 
and  Their  Relation  to  the  Transmission  System.  (Trans.  Am.  Inst. 
Elec.  Engrs.,  Vol.  18,  p.  371.) 

1902. 

ERNST  DANIELSON. — A  Novel  Combination  of  Polyphase  Motors  for 
Traction  Purposes.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  19,  p.  527.) 

CHAS.  P.  STEINMETZ. — Notes  on  the  Theory  of  the  Synchronous 
Motor.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  19,  p.  781.) 


1903. 

C.  A.  ADAMS. — A  Study  of  the  Heyland  Machine  as  Motor  and 
Generator.     (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  21,  p.  519.) 

H.  BEHN-ESCHENBURG. — Magnetic  Dispersion  in  Induction  Motors. 
(Jour.  Inst.  Elec.  Engrs.,  Vol.  33,  p.  239.) 

1904. 

B.  G.  LAMME. — Synchronous  Motors  for  Regulation  of  Power  Factor 
and  Line  Pressure.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  23,  p.  481.) 

H.  M.  HOBART. — The  Rated  Speed  of  Electric  Motors  as  Affecting 
the  Type  to  be  Employed.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  33,  p.  472.) 

1905. 

R.  GOLDSCHMIDT. — Temperature  Curves  and  the  Rating  of  Electrical 
Machinery.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  34,  p.  660.) 

D.  K.  MORRIS  and  G.  A.  LISTER. — Eddy-Current  Brake  for  Testing 
Motors.     (Jour.  Inst.  Elec.  Engrs.,  Vol.  35,  p.  445.) 

P.  D.  IONIDES. — Alternating-Current  Motors  in  Industrial  Service. 
(Jour.  Inst.  Elec.  Engrs.,  Vol.  35,  p.  475.) 


254       POLYPHASE  GENERATORS  AND  MOTORS 

C.  A.  ADAMS. — The  Design  of  Induction  Motors.  (Trans.  Am.  Inst. 
Elec.  Engrs.,  Vol.  24,  p.  649.) 

CHAS.  A.  PERKINS. — Notes  on  a  Simple  Device  for  Finding  the  Slip 
of  an  Induction  Motor.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  24,  p.  879.) 

A.  S.  LANGSDORF. — Air-Gap  Flux  in  Induction  Motors.  (Trans.  Am. 
Inst.  Elec.  Engrs.,  Vol.  24,  p.  919.) 

1906. 

\ 

J.  B.  TAYLOR.— Some  Features  Affecting  the  Parallel  Operation  of 
Synchronous  Motor-Generator  Sets.  (Trans.  Am.  Inst.  Elec.  Engrs. , 
Vol.  25,  p.  113.) 

BRADLEY  McCoRMiCK. — Comparison  of  Two-  and  Three-Phase 
Motors.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  25,  p.  295.) 

A.  BAKER  and  J.  T.  IRWIN. — Magnetic  Leakage  in  Induction  Motors. 
(Jour.  Inst.  Elec.  Engrs.,  Vol.  38,  p.  190.) 

1907. 

L.  J.  HUNT. — A  New  Type  of  Induction  Motor.  (Jour.  Inst.  Elec. 
Engrs.,  Vol.  39,  p.  648.) 

R.  RANKIN. — Induction  Motors.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  39, 
p.  714.) 

C.  A.  ADAMS,  W.  K.  CABOT,  and  C.  A.  IRVING,  Jr. — Fractional-Pitch 
Windings  for  Induction  Motors.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol. 
26,  p.  1485.) 

R.  E.  HELLMUND. — Zigzag  Leakage  of  Induction  Motors.  (Trans. 
Am.  Inst.  Elec.  Engrs.,  Vol.  26,  p.  1505.) 

1908. 

R.  GOLDSCHMIDT. — Standard  Performances  of  Electrical  Machinery. 
(Jour.  Inst.  Elec.  Engrs.,  Vol.  40,  p.  455.) 

G.  STEVENSON. — Polyphase  Induction  Motors.  (Jour.  Inst.  Elec. 
Engrs.,  Vol.  41,  p.  676.) 

H.  C.  SPECHT. — Induction  Motors  for  Multi-Speed  Service  with 
Particular  Reference  to  Cascade  Operation.  (Trans.  Am.  Inst.  Elec. 
Engrs.,  Vol.  27,  p.  1177.) 

1909. 

J.  C.  MACFARLANE  and  H.  BURGE. — Output  and  Economy  Limits  of 

Dynamo-Electric  Machinery.     (Jour.  Inst.  Elec.  Engrs.,  Vol.  42,  p.  232.) 

S.   B.   CHARTERS,   Jr.,   and  W.   A.   HILLEBRANDT. — Reduction   in 


APPENDIX  255 

Capacity  of  Polyphase  Motors  Due  to  Unbalancing  in  Voltage.  (Trans. 
Am.  Inst.  Elec.  Engrs.,  Vol.  28,  p.  559.) 

H.  G.  REIST  and  H.  MAXWELL. — Multi-Speed  Induction  Motors. 
(Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  28,  p.  601.) 

A.  MILLER  GRAY. — Heating  of  Induction  Motors.  (Trans.  Am.  Inst. 
Elec.  Engrs.,  Vol.  28,  p.  527.) 

1910. 

R.  E.  HELLMUND. — Graphical  Treatment  of  the  Zigzag  and  Slot 
Leakage  in  Induction  Motors.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  45,  p.  239.) 

C.  F.  SMITH. — Irregularities  in  the  Rotating  Field  of  the  Polyphase 
Induction  Motor.  (Jour.  Inst  Elec.  Engrs.,  Vol.  46,  p.  132.) 

WALTER  B.  N YE. —The  Requirements  for  an  Induction  Motor  from 
the  User's  Point  of  View.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  29, 
p.  147.) 

1911. 

T.  F.  WALL. — The  Development  of  the  Circle  Diagram  for  the 
Three-Phase  Induction  Machine.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  48, 
p.  499.) 

N.  PENSABENE-PEREZ. — An  Automatic  Starting  Device  for  Asyn- 
chronous Motors.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  48,  p.  484.) 

C.  F.  SMITH  and  E.  M.  JOHNSON — The  Losses  in  Induction  Motors 
Arising  from  Eccentricity  of  the  Rotor.  (Jour.  Inst.  Elec.  Engrs., 
Vol.  48,  p.  546.) 

H.  J.  S.  HEATHER. — Driving  of  Winding  Engines  by  Induction 
Motors.  (Jour.  Inst.  Elec.  Engrs.,  Vol.  47,  p.  609.) 

THEODORE  HOOCK. — Choice  of  Rotor  Diameter  and  Performance  of 
Polyphase  Induction  Motors.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  30.) 

Gus  A.  MAIER. — Methods  of  Varying  the  Speed  of  Alternating 
Current  Motors.  (Trans.  Am.  Inst.  Elec.  Engrs.,  Vol.  30.) 

1912. 

J.  K.  CATTERSON-SMITH. — Induction  Motor  Design.  (Jour.  Inst. 
Elec.  Engrs.,  Vol.  49,  p.  635.) 

CARL  J.  FECHHEIMER. — Self-Starting  Synchronous  Motors.  (Trans. 
Am.  Inst.  Elec.  Engrs.,  Vol.  31.) 

H.  C.  SPECHT. — Electric  Braking  of  Induction  Motors.  (Trans.  Am. 
Inst.  Elec.  Engrs.,  Vol.  31.) 

P.  M.  LINCOLN. — Motor  Starting  Currents  as  Affecting  Large  Trans- 
mission Systems,  (Trans.  Am,  Inst,  Elec,  Engrs.,  Vol.  31.) 


APPENDIX  III 


TABLE  OF  SINES,  COSINES,  AND  TANGENTS  FOR  USE  IN  THE  CALCULATIONS 

IN  CHAPTER  II. 


Angle. 

Sin. 

Cos. 

Tan. 

Angle. 

Sin. 

Cos. 

Tan. 

0° 

0 

1  .000 

0 

1° 

0.01745 

0.999 

0.01745 

46° 

0.719 

0.695 

1.035 

2° 

0.03490 

0.999 

0.03490 

47° 

0.731 

0.682 

1.072 

3° 

0.0523 

0,998 

0.0524 

48° 

0.743 

0.669 

1.111 

4° 

0.0697 

0.998 

0.0619 

49° 

0.755 

0.656 

1.150 

5° 

0.0871 

0.996 

0.0874 

50° 

0.766 

0.643 

1.192 

6° 

0.1045 

0.994 

0.1051 

51° 

0.777 

0.629 

1.235 

7° 

0.1218 

0.992 

0.1227 

52° 

0.788 

0.616 

1.280 

8° 

0.1391 

0.990 

0.1405 

53° 

0.799 

0.602 

1.327 

9° 

0.1564 

0.988 

0.1583 

54° 

0.809 

0.588 

1.376 

10° 

0.1736 

0.985 

0.1763 

55° 

0.819 

0.574 

1.428 

11° 

0.1908 

0.982 

0.1943 

56° 

0.829 

0.559 

1.482 

12° 

0.2079 

0.978 

0.2125 

57° 

0.839 

0.545 

1.540 

13° 

0.2249 

0.974 

0.2308 

58° 

0.848 

0.530 

1.600 

14° 

0.2419 

0.970 

0.2493 

59° 

0.857 

0.515 

1.664 

15° 

0.2588 

0.966 

0.2679 

60° 

0.866 

0.500 

1.732 

16° 

0.2756 

0.961 

0.2867 

61° 

0.875 

0.4848 

1.804 

17° 

0.2923 

0.956 

0.3057 

62° 

0.883 

0.4694 

1.880 

18° 

0.3090 

0.951 

0.3249 

63° 

'0.891 

0.4539 

1.963 

19° 

0.3255 

0.945 

0.3443 

64° 

0.899 

0.4383 

2.050 

20° 

0.3420 

0.940 

0.3639 

65° 

0.906 

0.4226 

2.144 

21° 

0.3583 

0.934 

0.3838 

66° 

0.914 

0.4067 

2.246 

22° 

0.3746 

0.927 

0.4040 

67° 

0.920 

0.3907 

2.356 

23° 

0.3907 

0.920 

0.4244 

68° 

0.927 

0.3746 

2.475 

24° 

0.4067 

0.914 

0.4452 

69° 

0.934 

0.3583 

2.605 

25° 

0.4226 

0.906 

0.4663 

70° 

0.940 

0.3420 

2.747 

26° 

0.4383 

0.899 

0.4877 

71° 

0.945 

0.3255 

2.904 

27° 

0.4539 

0.891 

0.509 

72° 

0.951 

0.3090 

3.077 

28° 

0.4694 

0.883 

0.532 

73° 

0.956 

0.2923 

3.271 

29° 

0.4848 

0.875 

0.554 

74° 

0.961 

0.2756 

3.487 

30° 

0.500 

0.866 

0.577 

75° 

0.966 

0.2588 

3.732 

31° 

0.515 

0.857 

0.601 

76° 

0.970 

0.2419 

4.011 

32° 

0.530 

0.848 

0.625 

77° 

0.974 

0.2249 

4.331 

33° 

0.545 

0.839 

0.649 

78° 

0.978 

0.2079 

4.705 

34° 

0.559 

0.829 

0.674 

79° 

0.982 

0.1908 

5.14 

35° 

0.574 

0.819 

0.700 

80° 

0.985 

0.1736 

5.67 

36° 

0.588 

0.809 

0.726 

81° 

0.988 

0.1564 

6.31 

37° 

0.602 

0.799 

0.754 

82° 

0.990 

0.1391 

7.11 

38° 

0.616 

0.788 

0.781 

83° 

C.992 

0.1218 

8.14 

39° 

0.629 

0.777 

0.810 

84° 

0.994 

0.1045 

9.51 

40° 

0.643 

0.766 

0.839 

85° 

0.996 

0.0871 

11.4 

41° 

0.656 

0.755 

0.869 

86° 

0.998 

0.0697 

14.3 

42° 

0.669 

0.743 

0.900 

87° 

0.998 

0.0523 

19.1 

43° 

0.682 

0.731 

0.932 

88° 

0.999 

0.03490 

28.6 

44° 

0.695 

0.719 

0.966 

89° 

0.999 

0.01745 

57.3 

45° 

0.707 

0.707 

1.000 

90° 

1.000 

0 

Infinite 

256 


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INDEX 


Air  Circulation,  see  Ventilation 
Air-gap 

Density,  Method  of  estimating  for 

Synchronous  Generators,  36 
Diameter,  Explanation  of  term,  4 
(of)  Induction  Generators,  217,  219, 

222-224 

(of)  Induction  Motors,  132-134 
Radiating  Surface  at  the,  Data  for 

estimating,  181 

(of)  Synchronous  Generator,  36,  37 

A.I.E.E.,  Bibliography  of  papers  on 

Polyphase    Generators,    246- 

251 — on  Polyphase  Motors,  252- 

255 

Apparent  Resistance  of  Squirrel-cage 

Rotor,  168 
Armature 

Inference,  34  et  seq;  43  et  seq;  57 — 

Relation  between  Theta  and,  52 

Magnetomotive  Force,  Axis  of,  45, 

46 

Reaction  with  Short-circuited  Arm- 
ature, 57 

Resistance,    Estimation    for    Syn- 
chronous Generator,  21 
Strength,   see   Armature   Interfer- 
ence 

Auto-transformer,  Connections  for 
starting  up  Induction  Motor  by 
means  of,  170 

Axis  of  Armature  Demagnetization, 
45,46 

Bibliography  of  Papers  on 
Circle  Ratio,  153 
Polyphase   Generators   246-251,— 

Polyphase  Motors,  252-255 
Breakdown     Factor     in     Induction 
Motors,  Determination  of,  186- 
191 


Calculations  for 

Slip  Ring  Induction  Motor,  195- 
201 

Squirrel-cage  Induction  Motor, 
105  et  seq — Design  of  Squirrel- 
cage  for,  173-178 

Synchronous  Generator,  3  et  seq — 
Specification  for,  40,  41 — Deriva- 
tion of  Design  for  Induction 
Generator  from,  216-222 
Circle  Diagram  of  Squirrel-cage 
Motor,  154,  170 

General  Observations  regarding, 
172 

Locus  of  Rotor  Current  in,  165 
Circle  Ratio, 

Bibliography  of,  Papers  on,  153 

(in)  Induction  Generator,  222 

Kierstead's  Formula,  152,  153 

(in)  Slip-ring  Induction  .Motor, 
200 

(in)   Squirrel-cage  Induction  Mo- 
tors, Estimation  of,  141  et  seq — 
Formula     for    estimating,    152, 
153 

Values  of,  Table,  150,  151 
Circulation  of  Air,  see  Ventilation 
Compensator 

Connections  for  starting  up  an 
Induction  Motor  by  means  of, 
170 

Step-up,  for  Induction  Generator, 

223 
Concentrated  Windings,  Inductance 

Calculations  in,  48,  49 
Conductors 

Copper,  Table  of  Properties  of, 
Appendix,  257 

Rotor  in  Induction  Generators, 
214,  215,  217-219— in  Induction 
Motors,  167 

259 


260 


INDEX 


Conductors — Continued 

(in)  Slip-ring  Induction  Motors, 
196-199 
(in)        Squirrel-cage        Induction 

Motors,  113-117,  127 
(in)  Synchronous  Generators,  9,  10 
Constant     Losses     in     Synchronous 
Generators,      94-96 — at      high 
speeds,  99-101 

Copper  Conductors,   Table  of  Pro- 
perties of,  Appendix,  257 
Core 

Densities  in  Induction  Motors,  135 
Length  in 

Induction  Generators,  218 
Induction  Motors,  108 
Synchronous  Generators,  6,  26 
Loss  in 

Induction  Motors,  Data  for  esti- 
mating, 158  et  seq 
Synchronous  Generators,  92,  99, 

100 

Cosines,  Table  of,  Appendix,  256 
Cost,  see  Total  Works  Cost  of  Induc- 
tion Motors 
Crest    Flux    Density    in    Induction 

Motors,  121 

Critical  Speed  of  Vibration,  224 
Current 

Density  suitable  for 

Induction  Motors,  127,  128 
Synchronous  Generators,  9,  10 
(in)  End  Rings  of  Squirrel- cage, 

176 

Ideal  Short-circuit,  154,  200 
Magnetizing  of  Induction  Motor, 

139,  140 
(at)  Rated  Load 
Squirrel-cage  Induction  Motor, 

Values  for  115,  116 
Synchronous  Generator,  Estima- 
tion, of,  8 
Curves 

Efficiency,  97,  181,  192,  193 
Excitation  Regulation,  71-73 
Power-factor,  192,  193 
Saturation,  74 — No-load  Satura- 
tion, 37-39 
(of)  Slip,  194 

Short-circuit  for  Synchronous  Gen- 
erators, 76-78 
Speed,  194 
Volt-ampere,  75,  83 


Diameters  of 

Squirrel-cage    Induction    Motors, 

136 

Synchronous  Generators,  Tabula- 
tion of  ,  28,  31,  37 

Distributed    Field    Windings,    Poly- 
phase Generators  with,  99  et  seq 
Dynamic   Induction,   Discussion  of, 
12,  13 

Eddy  Current  Losses  in   Rotor  Con- 
ductors as  affecting  the  Torque, 
168,  169,  177,  192,  194 
Efficiency  (of) 

Curves,  97,  181,  192,  193 

Dependence  of  on  Power-factor  of 
Load,  97,  101,  102 

Induction  Generator,  219 

Squirrel-cage  Induction  Motors, 
Methods  of  Calculating,  178  et 
seq.;  192,  193.  Values  for,  115, 
116 

Synchronous  Generators,  Methods 

of  Calculating,  95-98;  101,  102 
End  Rings  in 

Induction  Generators,  218,  219 

Squirrel-cage  Rotor,  174-176 — Use 
of  Magnetic  Material  for,  206, 
207 

Energy,  Motor  Transformer  of,  160 
Equivalent  Radiating  Surface  at  Air- 
gap,  Data  for  estimating,  181 
Equivalent  Resistanceof  Squirrel-cage 

Induction  Motor,  167,  168,  194 
Examples  for  Practice  in  Designing 
Polyphase  Generators  and  Mo- 
tors, 225-245 
Excitation 

(of)  Induction  Generators  supplied 
from  Synchronous  Apparatus  on 
System,  213 
Loss  in  Synchronous  Generators, 

93-96;  99,  100 

Pressure  for  Synchronous  Gener- 
ators, 84  et  seq. 
Regulation  Curves,  71-73 

Field,  A.  B.  on  Eddy  Current  Losses 
in  Copper  Conductors,  168,  210 

"Field  Effect"  for  improving  Start- 
ing Torque  in  Squirrel-cage  In- 
duction Motors,  168;  210-212 


INDEX 


261 


Field  Excitation,  45  et  seq.     Calcula- 
tions for  Synchronous  Generator, 
57,58 
Field  Spools,  Design  for  Synchronous 

Generator,  84-92 
Flux    per    Pole,    Estimation    of    in 

Synchronous  Generator,  18-22 
Formula 

Circle  Ratio,  Kierstead's  Formula 

for  estimating,  152,  153 
(for)    Current    in    End    Rings    of 

Squirrel  cage,  176 
(for)  Equivalent  Radiating  Surface 

at  Air-gap,  181 
Field  Effect,  168 
(for)  Mean  Length  of  Turn,  21, 

156,  199 

Output  Coefficient,  6,  7,  108 
Peripheral  Speed,  113 
Pressure 

Discussion  leading  up  to  Deriva- 
tion of,  13,  14 
(for)     Squirrel-cage     Induction 

Motors,  119 

Winding  Pitch  Factor  in,  16,  20 
Reactance,  48 
Theta,  51 

Total  Works  Cost,  111 
Two-circuit  Armature  Winding,  197 
Fractional  Pitch  Windings,  16-18 
Friction  Losses  in 

Induction  Motors,  Data  for  esti- 
mating, 166 
Synchronous  Generators,  93-96;  at 

high  speeds,  99,  100 
Full  Load  Power-factor,  Estimation 
of,    in    Squirrel-cage    Induction 
Motors,  155 
Full  Pitch  Windings,  14-18;  120,  197 

Generators 

Induction,  see  Induction  Genera- 
tors 
Synchronous,      see      Synchronous 

Generators 
Synchronous  versus  Induction,  209, 

210,  213 

Goldschmidt,  Dr.  Rudolf,  on  Power- 
factors  of  Induction  Motors,  202 
Gross  Core  Length  in 

Induction  Generators,  218 
Squirrel-cage  Induction  Motors,  108 
Synchronous  Generators,  6 


Half-coiled  Windings,  18-20;  41-43 

I.E.E.,  see  Institution  of  Electrical 

Engineers 

Ideal  Short-circuit  Current,  154;  200 
Impedance,  76,  77 

Inductance  of  Armature  Windings  of 
Synchronous  Generators,  45-50; 
53 
Induction  Generators,  Design  of,  1; 

213-224 

Derivation  of  Design  from  Design 
of  Synchronous  Generators,  216— 
222 

Speeds,    Appropriateness    for    ex- 
ceedingly high,  213 
Synchronous     Generators     versus, 

209,  210,  213 

Ventilating,  Methods  of,  220.   221 
Induction  Motors,  1 
Slip  Ring,  195-201 
Squirrel-cage,  Design  of,  105  et  seq. 
Magnetic   End   Rings,    Use   of, 

206,  207 

Open  Protected  Type,  186 
Slip  Ring,  Discussion  of  the  rela- 
tive   merits    of    Squirrel-cage 
and,  195-201 

Squirrel-cage  Design,  173-178 
Synchronous  Motors  versus,  202-212 
Inherent  Regulation,  44;  54;  55;  66; 

71;  80-82 

Institution  of  Electrical  Engineers, 
Bibliography  of  papers  on  Poly- 
phase  Generators,  246-251 — on 
Polyphase  Motors,  252-255 
Insulation  (of) 

Field  Spools,  88,  89 

Lamination  of  Induction  Motors, 

126 
Slot  (in) 

Slip-ring  Induction  Motors,  196; 

200 
Squirrel-cage  Induction  Motors, 

114;  128-130 
Synchronous  Generators,  10,  11 

Kierstead's  Formula  for  Circle  Ratio, 
152;  153 

Lap  Winding,  18;  19;  197 
Leakage  Factor,   22;    23;    42.     See 
also  Circle  Ratio 


262 


INDEX 


Losses  (in) 

Squirrel-cage    Induction    Motors, 

155  et  seq.;   166;  178  et  seq. 
Synchronous    Generator,    93-98 — 

Effect  of  High  Speed  on,  99-101 

Magnet  Core,  Material  and  Shape 
suitable  for  2500  kva.  Synchro- 
nous Generator,  23-25 

Magnet  Yoke,  Calculations  for  Syn- 
chronous Generator,  29 

Magnetic 
Circuit 

Squirrel-cage  Induction  Motors 
Design,  120  et  seq. 
Magnetomotive    Force,    Esti- 
mation of,  124;  134 
Mean  Length  of,  121 
Sketch,  137;  138 
Synchronous  Generators 
Design,  22  et  seq. 
Magnetomotive    Force,    Esti- 
mation of,  32  et  seq. 
Mean  Length  of,  31 
Data  for  Teeth  and  Air-gap  in  In- 
duction Motors,  134 
End  Rings,  Use  of  in  Squirrel-cage 

Motors,  206;  207 
Flux,  12  et  seq. 

Distribution  in  Induction  Mo- 
tors, 122-125 

Estimation  of  Flux  per  Pole,  for 
Synchronous  Generators,  18- 
22 

Materials,  Saturation  Data  of  va- 
rious, 32;   33 

Reluctance  of  Sheet  Steel,  137 
Magnetizing    Current    of   Induction 

Motor,  139;  140 
Magnetomotive  Force 
Axes  of  Field  and  Armature,  45; 

46 
Induction  Generator  Calculations, 

222 

Induction  Motor  Calculations,  134 
per    Cm.    for    various    Materials, 

Table,  33 

Synchronous    Generator    Calcula- 
tions, 32  et  seq,  54 
Tabulated  data  of,  32;  34;  38;  42; 
58;  61;  64;  65;  67;  69;  73;  74; 
137 


Mean  Length  of 

Magnetic     Circuit    in     Induction 
Motors,     121 — in     Synchronous 
Generators,  31 
Turn  of  Winding,  21 

in   Slip-ring   Induction   Motors, 

199 

in  Squirrel-cage  Induction  Mo- 
tors, 155-157 

Metric  Wire  Table,  Appendix,  257 
Motors,     Induction,     see    Induction 

Motors 
Motor  is  Transformer  of  Energy,  160 

Net    Core    Length    in    Synchronous 

Generators,  26 
No-load 

Current  of  Induction  Motor,  140 
Saturation      Curves,      37-39 — In- 
fluence of  Modifications  in,  78- 
83 

Open  Protected  Type  of  Squirrel-cage 

Induction  Motor,  186 
Output  Coefficient 

Formula,  Discussion  of  Significance 

of,  6;  7 
Squirrel-cage    Induction     Motors, 

Values  for,  108;  109 
Synchronous  Generator,   Table  of 

Values  for,  5;  6 

Output   from  Rotor   Conductors  in 
Induction  Motors,  167 

Partly    Distributed    Windings,    In- 
ductance Calculations,  48,  49 
Peripheral  Loading,  Appropriate  Val- 
ues f.or   Squirrel-cage   Induction 
Motors,     113-115 — for  Synchro- 
nous Generators,  8 
Peripheral  Speed  of 

Squirrel-cage     Induction    Motors, 

113 

Synchronous  Generators,  25 
Pitch 

Polar,  see  Polar  Pitch 
Rotor  Slot  in 

Slip-ring  Induction  Motors,  196 
Squirrel-cage  Induction  Motors, 
174;  196 

Slot,  see  Tooth  Pitch 
Tooth,  see  Tooth  Pitch 
(of)  Windings,  16-18;  120;  197 


INDEX 


263 


Polar  Pitch,  Suitable  Values  for 
Squirrel-cage    Induction     Motors, 

106;  107 

Synchronous  Generators,  4;  5 
Poles,  Data  for  number  of  in 

Squirrel-cage    Induction     Motors, 

106 

Synchronous  Generators,  3 
Power-factor 
Curves,  192;  193 
Efficiency,  Dependence  of  on  P.F. 

of  Load,  97;  101;  102 
Induction  Motors  versus  Synchron- 
ous Motors,  202-212 
Saturation  Curves,  Estimation  of 

for  various,  55  et  seq 
Squirrel-cage    Induction    Motors, 

Estimation  of,  115;    116;    155; 

191-193 
Pressure 
Formula 

Discussion  leading  up  to  deri- 
vation of,  13;  14 

(for)     Squirrel-cage     Induction 
Motors,  119 

Winding  Pitch  Factor  in,  16;  120 
Regulation,  Method  of  Estimating, 

39  et  seq 
Total    Internal    of    Synchronous 

Generator,  53 

Radial  Depth  of  Air-gap  (for) 
Induction  Motors,  133 
Synchronous  Generators,  36;  37 
Radiating  Surface  at  Air-gap,  Data 

for  Estimating,  181 
Ratio  of  Transformation  in 
Induction  Generators,  217;  223 
Slip-ring  Induction  Motors,  199 
Squirrel-cage    Induction     Motors, 

174 

Reactance  of  Windings  of  Synchro- 
nous Generators,  48 
Reactance  Voltage,  53 — Determina- 
tion of  Value  for  Synchronous 
Generators,  48-51 

Regulation,  Excitation  for  Synchro- 
nous Generators,  71-73 
Resistance    of    Squirrel-cage    Rotor, 

167;  168;  194 

Robinson,  L.  T.,  Skin  Effect  Investi- 
gation on  Machine-steel  Bars, 
208;  209 


Rotor  (of) 

Induction  Generators,  223;  224 
C&nductors  in,  214  et  seq 
Slots  in,  217 

Slip-ring  Induction  Motor, 
Slot  Pitch,  196 
Slots,  195-197 
Windings  for,  195;   197-199 
Squirrel-cage  Induction  Motor 
Conductors, 

Eddy  Current  Losses  as  affect- 
ing the  Torque,    168;  169; 
177;  192;  194 
Output  from,  167 
Core 

Densities  in,  135 

Loss  in,  158;    159;    164-166; 

193;  194 

Material  for,  Choice  of;   158 
Resistance,  167;  168 
Slots 

Design  of,  132 
Number,  173 
Pitch,  174;  196 

Squirrel-cage,    Design    of,    173- 
178 

Salient  Pole  Generator,  Calculations 

for  2500  kva.,  3  et  seq 
Derivation  of  Design  for  Induction 

Generator  from;   216-222 
Specification  of,  40 
see  also  Synchronous  Generator 
Saturation 

Curves  for  Synchronous  Generator, 
55  et  seq;  74 

No-load    Curves,    37-39— Influ- 
ence of  Modifications  of,  78-83 
Data  of  various  Magnetic  Mate- 
rials, 32;  33 
Sheet  Steel,  Magnetic  Reluctance  of 

137 
Short-circuit  Curve  for  Synchronous 

Generators,  76-78 
Sines,  Table  of,  Appendix,  256 
Single-layer  Windings,  18 
Skin    Effect    to    improve    Starting 
Torque  of  Synchronous  Motors, 
207-209 

Slip,  106;  163;  164;  192;  194;  215 
Slip-ring   Induction   Motor,    Discus- 
sion of  relative  merits  of  Squirrel- 
cage  and,  195-201 


264 


INDEX 


Slot-embedded  Windings,  Inductance 

and  Reactance  of,  48-50 
Slot 

Insulation  in 

Induction    Motors,    114;     128- 

130;  196;  200 

Synchronous  Generator,  10;  11 
Pitch,  see  Tooth  Pitch 
Space  Factor,  11;  131 
Tolerance,  127;  196 
Slots 

Rotor,  for 

Induction  Generators,  217 
Slip-ring  Induction  Motors,  195- 

197 
Squirrel-cage  Induction  Motors, 

132;  141;  173;  174;  196 
Stator,  for 

Induction  Generators,  217 
Induction  Motors,  117-119;  127; 

131;  141 

Synchronous  Generators,  9;  11 
Space  Factor  (of) 

Field  Spools  of  Synchronous  Gen- 
erators, 88;  89 
Slot,  11;  131 

Specification  of  2500  kva.  Synchro- 
nous Generator,  40;  41 
Speed 

Control,    Methods    of     providing, 

195 

Curves,  194 
High-speed    Sets,    Characteristics 

of,  99 

Synchronous  versus  Induction  Mo- 
tors for  Low  and  High,  202-212 
Spiral  Windings,  18-20 
Spread  of  Winding,  14-16;  120 
Spreading  Coefficients,  36 
Squirrel-cage    Induction.  Motor,    see 

Induction  Motor 
Starting     Torque     of     Squirrel-cage 

Motor,  169-172 
Stator 

Conductors,  Determination  of  Di- 
mensions for  Induction  Motors, 
127 
Core  of 

Induction  Motors 
Density,  135 
Loss,  158;  159 
Material  preferable  for,  158 
Weight  of,  Estimation,  159 


Stator — Continued 
Core  of — Continued 

Synchronous  Generators,  27;  28 

Weight,  92 

Current  Density  suitable  for  Induc- 
tion Motors,  127;  128 
PR  Loss  in 

Induction  Motors,  155-157 
Synchronous  Generators,  93-96 
Slot  Pitch,   Values  for  Induction 

Motors,  118 
Slots  in 
»        Induction  Generators,  217 

Induction    Motors,     119;     127; 

131 

Synchronous  Generators,  11 
Teeth,  Data  for  Induction  Motors, 

121-127 

Steam-turbine   Driven   Sets,   Rotors 
with  Distributed  Field  Windings 
for,  99 — Circulating  Air  Calcu- 
lations, 102-104 
Step-down    Transformers    for    small 

Motors,  114 
Step-up  Transformer  for  Induction 

Generator,  223 
Synchronous  Generators,  1 

Distributed  Field  Winding  Type, 

99  et  seq 

Efficiency,     Dependence     of     on 
Power-factor  of  Load,  97;    101; 
102 
Induction   Generator  versus,   209; 

210;  213 
Salient  Pole  Type 

Calculations  for  2500  kva.,  3  et 

seq 

Derivation  of  Design  for  Induc- 
tion    Generator     from,    216- 

222 

Specification,  40;  41 
Synchronous    Motors,    1 — Induction 
Motors  versus,  202-212 

Tabulated  Data  of  Magnetomotive 
Force  Calculations,  32;   34;  38; 
42;  58;  61;  64;  65;  67;  69;  73; 
74;  137 
Tabulation  of 

Losses  and  Efficiencies  in  Squirrel- 
cage  Induction  Motors,  180 
Squirrel-cage  Induction  Motor  Di- 
ameters, 136 


INDEX 


265 


Tabulation  of — Continued 

Synchronous  Generator  Diameters, 

28,  31:  37 

Tangents,  Table  of,  Appendix,  256 
Teeth,  Stator,  in  Induction  Motors, 

121-127 

Temperature  Rise,  Data  for  estima- 
ting, 90;  181-183 

Theta  and  its  Significance,  51  et  seq 
Thoroughly    Distributed    Windings, 
Inductance  Calculations,  48;  49 
Tooth 

Densities    in    Induction    Motors, 

121-125 

Pitch  26;  118;  196 
Torque,  162;  195 

Eddy    Current    Losses    in    Rotor 
Conductors    as    affecting,    168; 
169;   177;   192;   194 
Starting  (of) 

Squirrel-cage  Induction    Motor, 

169-172 

Synchronous    Motors    appropri- 
ate for  high,  204  et  seq 
Torque  Factor,  167 
Total  Net  Weight  of 

Induction  Motors,  110;  111 — Data 

for  estimating,  185 
Synchronous  Generators,  7 
Total    Works     Cost     of     Induction 
Motors,   158 — Methods  of  esti- 
mating, 111-113;  118 
Transformers 

Auto,  for   starting    up    Induction 

Motors,  170 

Step-down,  for  small  Motors,  114 
Step-up,  for  Induction  Generators, 

223 
Two-layer  Winding,  18;  197 


Variable  Losses  in  Synchronous  Gen- 
erators, 94-96 
Ventilating  Ducts  for 

Induction  Motors,  125;  126 
Synchronous  Generators,  25;  26 
Ventilation  of 

Induction     Generators,     Methods 

suitable  for,  220;  221 
Synchronous  Generators  with  Dis- 
tributed Field  Windings,  102-104 
Vibration,  Critical  Speed  of,  224 
Volt-ampere  Curves,  75,  83 
Voltage 

Formula,  see  Pressure  Formula 
Regulation,  Method  of  Estimating, 
39  et  seq 

Watts    per    Ton    for    Squirrel-cage 

Induction  Motor,  183-186 
Weight  of 

Induction  Motors,  110;  111;  183- 

185 
Stator  Core  of  Induction  Motors, 

159 

Synchronous  Generators,  7 
Whole-coiled  Windings,  18-20 
Winding 

Pitch,  16-18;  120;  197 
Pitch  Factor,  16;  120 
Spreads,  14-16;  120 
Inductance  and  Reactance  of,  45- 

49 
Types  of,  16-20;  41-43 

Distributed     Field,     Polyphase 

Generators  with,  99  et  seq 
(for)  Slip-ring  Induction  Motors, 

197-199 
Wire  Table,  Metric,  Appendix,  257   • 


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